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Properties

Label 2-624-208.19-c1-0-30
Degree $2$
Conductor $624$
Sign $0.826 + 0.562i$
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  + (−1.11 − 0.874i)2-s + (0.258 + 0.965i)3-s + (0.471 + 1.94i)4-s − 0.00444i·5-s + (0.556 − 1.29i)6-s + (0.544 − 2.03i)7-s + (1.17 − 2.57i)8-s + (−0.866 + 0.499i)9-s + (−0.00388 + 0.00494i)10-s + (2.11 + 1.21i)11-s + (−1.75 + 0.958i)12-s + (2.15 − 2.88i)13-s + (−2.38 + 1.78i)14-s + (0.00429 − 0.00115i)15-s + (−3.55 + 1.83i)16-s + (−1.42 + 0.824i)17-s + ⋯
L(s)  = 1  + (−0.785 − 0.618i)2-s + (0.149 + 0.557i)3-s + (0.235 + 0.971i)4-s − 0.00198i·5-s + (0.227 − 0.530i)6-s + (0.205 − 0.768i)7-s + (0.415 − 0.909i)8-s + (−0.288 + 0.166i)9-s + (−0.00122 + 0.00156i)10-s + (0.637 + 0.367i)11-s + (−0.506 + 0.276i)12-s + (0.598 − 0.801i)13-s + (−0.636 + 0.476i)14-s + (0.00110 − 0.000297i)15-s + (−0.889 + 0.457i)16-s + (−0.346 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.826 + 0.562i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08015 - 0.332669i\)
\(L(\frac12)\) \(\approx\) \(1.08015 - 0.332669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.874i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (-2.15 + 2.88i)T \)
good5 \( 1 + 0.00444iT - 5T^{2} \)
7 \( 1 + (-0.544 + 2.03i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.11 - 1.21i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.42 - 0.824i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.96 + 1.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.95 + 4.01i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.01 - 0.272i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-4.51 - 4.51i)T + 31iT^{2} \)
37 \( 1 + (-0.723 + 1.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-11.7 + 3.14i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-11.4 - 3.06i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.66 - 1.66i)T - 47iT^{2} \)
53 \( 1 + (-6.37 + 6.37i)T - 53iT^{2} \)
59 \( 1 + (8.62 - 4.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.36 + 8.84i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.8 + 7.41i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.51 - 1.74i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.07 - 4.07i)T - 73iT^{2} \)
79 \( 1 - 5.60iT - 79T^{2} \)
83 \( 1 + 3.12iT - 83T^{2} \)
89 \( 1 + (-2.19 - 8.17i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.02 - 1.07i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.65076376744554849942947543555, −9.711964814847705453467048677098, −8.943746946912911185549043456110, −8.090930624808371413441033461552, −7.26793098949486293351811419287, −6.15558076447507850961525750851, −4.54842420218290472343824121828, −3.80641141976726569790789954851, −2.61552032606948463561053671907, −0.987959036928981712760513875376, 1.24726045344292857021312124921, 2.47650639728101110443701641883, 4.22768356557772191550900021341, 5.74929542411316342603760307724, 6.18247934684179987862904145762, 7.28914459429507779202861375271, 8.054445483044178505847717860610, 8.996427056594508054554064937433, 9.371644765247403538108021317895, 10.62312981024082793672064518215

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