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Properties

Label 2-3e2-9.7-c3-0-1
Degree $2$
Conductor $9$
Sign $0.972 + 0.234i$
Analytic cond. $0.531017$
Root an. cond. $0.728709$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 − 1.18i)2-s + (−5.05 + 1.18i)3-s + (3.05 + 5.29i)4-s + (−5.18 − 8.98i)5-s + (−2.05 + 6.82i)6-s + (2.55 − 4.43i)7-s + 19.3·8-s + (24.1 − 12.0i)9-s − 14.2·10-s + (−27.9 + 48.4i)11-s + (−21.7 − 23.1i)12-s + (−18.7 − 32.5i)13-s + (−3.51 − 6.08i)14-s + (36.9 + 39.2i)15-s + (−11.1 + 19.3i)16-s + 23.6·17-s + ⋯
L(s)  = 1  + (0.242 − 0.420i)2-s + (−0.973 + 0.228i)3-s + (0.382 + 0.662i)4-s + (−0.463 − 0.803i)5-s + (−0.140 + 0.464i)6-s + (0.138 − 0.239i)7-s + 0.856·8-s + (0.895 − 0.445i)9-s − 0.450·10-s + (−0.767 + 1.32i)11-s + (−0.523 − 0.557i)12-s + (−0.400 − 0.694i)13-s + (−0.0670 − 0.116i)14-s + (0.635 + 0.676i)15-s + (−0.174 + 0.302i)16-s + 0.337·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(0.531017\)
Root analytic conductor: \(0.728709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :3/2),\ 0.972 + 0.234i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.768635 - 0.0913638i\)
\(L(\frac12)\) \(\approx\) \(0.768635 - 0.0913638i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.05 - 1.18i)T \)
good2 \( 1 + (-0.686 + 1.18i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (5.18 + 8.98i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-2.55 + 4.43i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (27.9 - 48.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (18.7 + 32.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 23.6T + 4.91e3T^{2} \)
19 \( 1 - 39.0T + 6.85e3T^{2} \)
23 \( 1 + (35.5 + 61.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-14.1 + 24.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (6.44 + 11.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 180.T + 5.06e4T^{2} \)
41 \( 1 + (-107. - 186. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (30.6 - 53.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-30.9 + 53.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 492.T + 1.48e5T^{2} \)
59 \( 1 + (394. + 683. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (260. - 451. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (152. + 263. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 270.T + 3.57e5T^{2} \)
73 \( 1 + 925.T + 3.89e5T^{2} \)
79 \( 1 + (-644. + 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (356. - 618. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 404.T + 7.04e5T^{2} \)
97 \( 1 + (37.5 - 64.9i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.92024990564730087997468083631, −20.12680868104661593832197230863, −17.87298242476415937223572926562, −16.70236525143043895601856729908, −15.57137665890157673680315591470, −12.79102408508266130325545875572, −12.02983002008330811332268236047, −10.35636168919633159210433083573, −7.57820955445575704468383890105, −4.67133977662776025619376926210, 5.61571502902450384024731321480, 7.23425613183800289541388769024, 10.55020127415590808373153523460, 11.63045977622110454064495631380, 13.86509770949156527156030336576, 15.45793030552983775794686496833, 16.53500639267541014152739849537, 18.45578351432266860477168241846, 19.27743361342803593284221203997, 21.48563163867450645685248518776

Graph of the $Z$-function along the critical line