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Properties

Label 2-2e9-64.21-c1-0-10
Degree $2$
Conductor $512$
Sign $0.996 + 0.0786i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.152 + 0.769i)3-s + (2.78 − 1.86i)5-s + (3.13 + 1.29i)7-s + (2.20 − 0.912i)9-s + (−1.79 − 0.356i)11-s + (−4.34 − 2.90i)13-s + (1.85 + 1.85i)15-s + (−1.16 + 1.16i)17-s + (−0.00265 + 0.00396i)19-s + (−0.519 + 2.61i)21-s + (−1.78 − 4.32i)23-s + (2.38 − 5.76i)25-s + (2.34 + 3.51i)27-s + (1.70 − 0.338i)29-s + 9.42i·31-s + ⋯
L(s)  = 1  + (0.0883 + 0.443i)3-s + (1.24 − 0.832i)5-s + (1.18 + 0.491i)7-s + (0.734 − 0.304i)9-s + (−0.540 − 0.107i)11-s + (−1.20 − 0.805i)13-s + (0.479 + 0.479i)15-s + (−0.281 + 0.281i)17-s + (−0.000608 + 0.000910i)19-s + (−0.113 + 0.569i)21-s + (−0.373 − 0.900i)23-s + (0.477 − 1.15i)25-s + (0.451 + 0.675i)27-s + (0.316 − 0.0628i)29-s + 1.69i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.996 + 0.0786i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.996 + 0.0786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96966 - 0.0775366i\)
\(L(\frac12)\) \(\approx\) \(1.96966 - 0.0775366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.152 - 0.769i)T + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (-2.78 + 1.86i)T + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (-3.13 - 1.29i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.79 + 0.356i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (4.34 + 2.90i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + (1.16 - 1.16i)T - 17iT^{2} \)
19 \( 1 + (0.00265 - 0.00396i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.78 + 4.32i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.70 + 0.338i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 - 9.42iT - 31T^{2} \)
37 \( 1 + (-1.47 - 2.20i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.497 - 1.20i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.104 + 0.523i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 + (-0.378 + 0.378i)T - 47iT^{2} \)
53 \( 1 + (-3.63 - 0.723i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (11.1 - 7.47i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.29 - 6.52i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-1.46 - 7.38i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (5.03 + 2.08i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (4.53 - 1.87i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (11.3 + 11.3i)T + 79iT^{2} \)
83 \( 1 + (-3.35 + 5.01i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (-4.25 + 10.2i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−10.42544323818889488653830897075, −10.19204259023641227232469102011, −9.066041293428415745346733534706, −8.459983097659106206228222404796, −7.36217230600473754522635608671, −5.99055942052353480024183584934, −5.04571358323777667216155785911, −4.59880575062560052685913024257, −2.66087958977400735403415299133, −1.47632178307931865714014140217, 1.76236577269754885469520400104, 2.43594967685220953150121209279, 4.33927866102995708457494044160, 5.26406055682966310700454733804, 6.45349047530001425515682106189, 7.36092553688641404476401199303, 7.86101786164845999528095749330, 9.420666712661488252317842806103, 9.998865014670008829871078342907, 10.84386615878870419370689155876

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