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Properties

Label 2-921-921.638-c0-0-0
Degree $2$
Conductor $921$
Sign $0.479 + 0.877i$
Analytic cond. $0.459638$
Root an. cond. $0.677966$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.602 − 0.798i)3-s + (−0.850 + 0.526i)4-s + (0.0170 + 0.183i)7-s + (−0.273 + 0.961i)9-s + (0.932 + 0.361i)12-s + (0.831 − 1.66i)13-s + (0.445 − 0.895i)16-s + (0.658 − 1.32i)19-s + (0.136 − 0.124i)21-s + (0.445 + 0.895i)25-s + (0.932 − 0.361i)27-s + (−0.111 − 0.147i)28-s + (−0.111 + 0.147i)31-s + (−0.273 − 0.961i)36-s + (−0.156 − 1.69i)37-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)3-s + (−0.850 + 0.526i)4-s + (0.0170 + 0.183i)7-s + (−0.273 + 0.961i)9-s + (0.932 + 0.361i)12-s + (0.831 − 1.66i)13-s + (0.445 − 0.895i)16-s + (0.658 − 1.32i)19-s + (0.136 − 0.124i)21-s + (0.445 + 0.895i)25-s + (0.932 − 0.361i)27-s + (−0.111 − 0.147i)28-s + (−0.111 + 0.147i)31-s + (−0.273 − 0.961i)36-s + (−0.156 − 1.69i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ 0.479 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6655293394\)
\(L(\frac12)\) \(\approx\) \(0.6655293394\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.602 + 0.798i)T \)
307 \( 1 + (0.602 + 0.798i)T \)
good2 \( 1 + (0.850 - 0.526i)T^{2} \)
5 \( 1 + (-0.445 - 0.895i)T^{2} \)
7 \( 1 + (-0.0170 - 0.183i)T + (-0.982 + 0.183i)T^{2} \)
11 \( 1 + (0.273 + 0.961i)T^{2} \)
13 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.658 + 1.32i)T + (-0.602 - 0.798i)T^{2} \)
23 \( 1 + (0.850 + 0.526i)T^{2} \)
29 \( 1 + (0.850 - 0.526i)T^{2} \)
31 \( 1 + (0.111 - 0.147i)T + (-0.273 - 0.961i)T^{2} \)
37 \( 1 + (0.156 + 1.69i)T + (-0.982 + 0.183i)T^{2} \)
41 \( 1 + (0.982 + 0.183i)T^{2} \)
43 \( 1 + (1.25 + 1.14i)T + (0.0922 + 0.995i)T^{2} \)
47 \( 1 + (0.602 + 0.798i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.932 + 0.361i)T^{2} \)
61 \( 1 + (-1.37 - 1.25i)T + (0.0922 + 0.995i)T^{2} \)
67 \( 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)T^{2} \)
71 \( 1 + (-0.0922 + 0.995i)T^{2} \)
73 \( 1 + (-0.0822 + 0.165i)T + (-0.602 - 0.798i)T^{2} \)
79 \( 1 + (0.890 - 0.811i)T + (0.0922 - 0.995i)T^{2} \)
83 \( 1 + (0.982 + 0.183i)T^{2} \)
89 \( 1 + (-0.739 - 0.673i)T^{2} \)
97 \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \)
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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

−10.26850511628060798520117578281, −9.074356681709589679283291244460, −8.439342947720130550229803138987, −7.59405711452866119821891565690, −6.89154567614638729029560338365, −5.47688476133510312363390580088, −5.26417375054384314439983255936, −3.76288109773803913358609315108, −2.69759968033241530113546341912, −0.840905234249990002817086923116, 1.39931044154617869889039252766, 3.52603841491770123383411079419, 4.29756214175514487991598596445, 5.03621352876476410582367120976, 6.06368322095144840088418278443, 6.66580162955427940002454616918, 8.226754414278374938076665939535, 8.909562367298117880634961771050, 9.805089370194961075051076505449, 10.16532528336911739608481998541

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