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Properties

Label 2-3e2-9.2-c2-0-0
Degree $2$
Conductor $9$
Sign $0.984 + 0.173i$
Analytic cond. $0.245232$
Root an. cond. $0.495209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)2-s + (−1.5 + 2.59i)3-s + (−0.5 − 0.866i)4-s + (3 − 1.73i)5-s + (4.5 − 2.59i)6-s + (−1 + 1.73i)7-s + 8.66i·8-s + (−4.5 − 7.79i)9-s − 6·10-s + (−1.5 − 0.866i)11-s + 2.99·12-s + (2 + 3.46i)13-s + (3 − 1.73i)14-s + 10.3i·15-s + (5.5 − 9.52i)16-s − 15.5i·17-s + ⋯
L(s)  = 1  + (−0.750 − 0.433i)2-s + (−0.5 + 0.866i)3-s + (−0.125 − 0.216i)4-s + (0.600 − 0.346i)5-s + (0.750 − 0.433i)6-s + (−0.142 + 0.247i)7-s + 1.08i·8-s + (−0.5 − 0.866i)9-s − 0.600·10-s + (−0.136 − 0.0787i)11-s + 0.249·12-s + (0.153 + 0.266i)13-s + (0.214 − 0.123i)14-s + 0.692i·15-s + (0.343 − 0.595i)16-s − 0.916i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(0.245232\)
Root analytic conductor: \(0.495209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :1),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.437343 - 0.0382626i\)
\(L(\frac12)\) \(\approx\) \(0.437343 - 0.0382626i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-3 + 1.73i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + (24 - 13.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-39 - 22.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (16 + 27.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 34T + 1.36e3T^{2} \)
41 \( 1 + (10.5 - 6.06i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 + 52.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (42 + 24.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (-43.5 + 25.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (28 - 48.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (19 - 32.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (42 + 24.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 124. iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 + 99.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−21.15948344451334266765784998646, −20.05760418643980437206185983639, −18.36252858369717876225436281035, −17.29822143466663229088948166946, −15.86169059689982381326924536166, −14.03827374318429537277580525527, −11.66600372164382799443895964574, −10.11159710619766847076183869512, −9.063973502092468953165449586453, −5.43739916243858562226877482479, 6.45974206135036256963934257592, 8.114120071487304495902274213765, 10.25834350509717015183038832534, 12.44252633372168062511752559250, 13.83133034304262492305122897953, 16.17910313472693676683295790448, 17.52416778794563222351021452872, 18.15348690502951163348951181700, 19.52798184900777749165917720555, 21.65185461246845770886572857494

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