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Properties

Label 2-624-13.3-c3-0-17
Degree $2$
Conductor $624$
Sign $-0.0942 - 0.995i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
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  + (1.5 + 2.59i)3-s + 15.3·5-s + (1.63 − 2.82i)7-s + (−4.5 + 7.79i)9-s + (−24.6 − 42.7i)11-s + (−8.10 + 46.1i)13-s + (23.0 + 39.8i)15-s + (−53.6 + 92.8i)17-s + (−59.9 + 103. i)19-s + 9.80·21-s + (15.2 + 26.3i)23-s + 110.·25-s − 27·27-s + (86.5 + 149. i)29-s + 189.·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 1.37·5-s + (0.0882 − 0.152i)7-s + (−0.166 + 0.288i)9-s + (−0.676 − 1.17i)11-s + (−0.172 + 0.984i)13-s + (0.396 + 0.686i)15-s + (−0.764 + 1.32i)17-s + (−0.724 + 1.25i)19-s + 0.101·21-s + (0.137 + 0.239i)23-s + 0.886·25-s − 0.192·27-s + (0.554 + 0.960i)29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.0942 - 0.995i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.0942 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.327506268\)
\(L(\frac12)\) \(\approx\) \(2.327506268\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 - 2.59i)T \)
13 \( 1 + (8.10 - 46.1i)T \)
good5 \( 1 - 15.3T + 125T^{2} \)
7 \( 1 + (-1.63 + 2.82i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (24.6 + 42.7i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (53.6 - 92.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (59.9 - 103. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-15.2 - 26.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-86.5 - 149. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 189.T + 2.97e4T^{2} \)
37 \( 1 + (-155. - 269. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (65.2 + 113. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (21.4 - 37.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 336.T + 1.03e5T^{2} \)
53 \( 1 - 673.T + 1.48e5T^{2} \)
59 \( 1 + (83.2 - 144. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-268. + 464. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (233. + 405. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (360. - 624. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 408.T + 3.89e5T^{2} \)
79 \( 1 - 1.09e3T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + (683. + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (559. - 969. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.41029735119839418695113896152, −9.662009468330108467508709885226, −8.690690594564922188174885667177, −8.154202135369727532309701295156, −6.54368026612706708342459600088, −5.99195759646267567917338731386, −4.94892576861295723098000931081, −3.82475573400228393341752659675, −2.55331493437110696052465700136, −1.51795067716779796728333925390, 0.61094149988249476526194285577, 2.35293768778400950626149670103, 2.52378132460520245529355317828, 4.60932640433738921532210191719, 5.38349497249315117780707349978, 6.47971892412206365843817733264, 7.19925912388504837367840658868, 8.245307177014464091164853418495, 9.230892038629014049169721176401, 9.884089950398394992790986458618

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