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Properties

Label 2-624-39.29-c0-0-0
Degree $2$
Conductor $624$
Sign $0.252 - 0.967i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (1 − 1.73i)19-s − 0.999·21-s + 25-s − 0.999·27-s + 31-s + (−1 − 1.73i)37-s − 0.999·39-s + (−0.5 + 0.866i)43-s + 1.99·57-s + (0.5 − 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (1 − 1.73i)19-s − 0.999·21-s + 25-s − 0.999·27-s + 31-s + (−1 − 1.73i)37-s − 0.999·39-s + (−0.5 + 0.866i)43-s + 1.99·57-s + (0.5 − 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000352109\)
\(L(\frac12)\) \(\approx\) \(1.000352109\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 - T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.97129559742215482040433422081, −9.931116091747346110423443473801, −9.161734658914559992170500016083, −8.819517375467821617261056172790, −7.52829590889521030325909695006, −6.54061499592056824897355517689, −5.28998269265951728654071309050, −4.58052234750667136602277376957, −3.23304083245014877016300954431, −2.39082741539602521707451450714, 1.23506609194945257266445349374, 2.88839019527895316305330121017, 3.71554517260195559653698089902, 5.24019114261790407986474765330, 6.35739422124495901714742421690, 7.18696686612219393493168256722, 7.905440660939254498211302819696, 8.706343166259501100413986556692, 9.981085007977267980040519429716, 10.32018202553473976428829763955

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