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Properties

Label 2-624-39.32-c1-0-4
Degree $2$
Conductor $624$
Sign $0.582 - 0.812i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
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  + (−1.19 − 1.25i)3-s + (2.36 + 2.36i)5-s + (0.332 − 1.24i)7-s + (−0.164 + 2.99i)9-s + (1.52 + 5.69i)11-s + (−3.51 + 0.821i)13-s + (0.159 − 5.79i)15-s + (−1.04 − 1.80i)17-s + (−1.34 − 0.360i)19-s + (−1.95 + 1.05i)21-s + (−3.81 + 6.60i)23-s + 6.20i·25-s + (3.96 − 3.35i)27-s + (6.87 + 3.96i)29-s + (−1.42 + 1.42i)31-s + ⋯
L(s)  = 1  + (−0.687 − 0.726i)3-s + (1.05 + 1.05i)5-s + (0.125 − 0.468i)7-s + (−0.0549 + 0.998i)9-s + (0.460 + 1.71i)11-s + (−0.973 + 0.227i)13-s + (0.0411 − 1.49i)15-s + (−0.252 − 0.436i)17-s + (−0.308 − 0.0827i)19-s + (−0.426 + 0.230i)21-s + (−0.795 + 1.37i)23-s + 1.24i·25-s + (0.762 − 0.646i)27-s + (1.27 + 0.736i)29-s + (−0.256 + 0.256i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.582 - 0.812i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.582 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12634 + 0.578612i\)
\(L(\frac12)\) \(\approx\) \(1.12634 + 0.578612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.19 + 1.25i)T \)
13 \( 1 + (3.51 - 0.821i)T \)
good5 \( 1 + (-2.36 - 2.36i)T + 5iT^{2} \)
7 \( 1 + (-0.332 + 1.24i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.52 - 5.69i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.04 + 1.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.34 + 0.360i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.81 - 6.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.87 - 3.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.42 - 1.42i)T - 31iT^{2} \)
37 \( 1 + (-4.28 + 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.985 - 0.263i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.68 + 5.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 - 2.59i)T - 47iT^{2} \)
53 \( 1 + 7.08iT - 53T^{2} \)
59 \( 1 + (1.48 + 0.397i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.39 - 5.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.22 - 8.30i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.21 + 4.53i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.61 + 9.61i)T + 73iT^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + (-0.406 - 0.406i)T + 83iT^{2} \)
89 \( 1 + (1.75 + 6.55i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.89 + 1.57i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.60178458275083528833568740644, −10.05902837013666546766116550434, −9.304493291393855379400727488903, −7.60378162985978932403790357333, −7.08501436371820022352838141364, −6.47425639240182627738698218756, −5.39149623666039758825916270402, −4.38903602573313487821310724505, −2.54068260498138317310969644288, −1.70897093368255333378680373097, 0.77185123035687727569677978876, 2.54162178922358166251144268119, 4.14316617448660440155781288305, 5.03245398277100134276940912527, 5.94270815444800062122789171125, 6.31592531746922289814617736829, 8.247173454676504903158663860357, 8.847691105482607197112406064699, 9.638068978258110970704502189801, 10.38408861334282134784494927459

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