Properties

Label 2-624-208.187-c1-0-22
Degree $2$
Conductor $624$
Sign $-0.562 + 0.826i$
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  + (−0.887 − 1.10i)2-s + (−0.707 − 0.707i)3-s + (−0.425 + 1.95i)4-s − 3.47·5-s + (−0.151 + 1.40i)6-s + (3.04 + 3.04i)7-s + (2.52 − 1.26i)8-s + 1.00i·9-s + (3.08 + 3.82i)10-s − 2.48·11-s + (1.68 − 1.08i)12-s + (−2.36 − 2.72i)13-s + (0.651 − 6.05i)14-s + (2.45 + 2.45i)15-s + (−3.63 − 1.66i)16-s − 0.873i·17-s + ⋯
L(s)  = 1  + (−0.627 − 0.778i)2-s + (−0.408 − 0.408i)3-s + (−0.212 + 0.977i)4-s − 1.55·5-s + (−0.0617 + 0.574i)6-s + (1.15 + 1.15i)7-s + (0.894 − 0.447i)8-s + 0.333i·9-s + (0.974 + 1.20i)10-s − 0.749·11-s + (0.485 − 0.312i)12-s + (−0.655 − 0.755i)13-s + (0.174 − 1.61i)14-s + (0.634 + 0.634i)15-s + (−0.909 − 0.415i)16-s − 0.211i·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.562 + 0.826i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246595 - 0.466195i\)
\(L(\frac12)\) \(\approx\) \(0.246595 - 0.466195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.887 + 1.10i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (2.36 + 2.72i)T \)
good5 \( 1 + 3.47T + 5T^{2} \)
7 \( 1 + (-3.04 - 3.04i)T + 7iT^{2} \)
11 \( 1 + 2.48T + 11T^{2} \)
17 \( 1 + 0.873iT - 17T^{2} \)
19 \( 1 - 8.32T + 19T^{2} \)
23 \( 1 + 2.16iT - 23T^{2} \)
29 \( 1 + (-2.02 + 2.02i)T - 29iT^{2} \)
31 \( 1 + (7.24 + 7.24i)T + 31iT^{2} \)
37 \( 1 + 4.91iT - 37T^{2} \)
41 \( 1 + (-4.08 - 4.08i)T + 41iT^{2} \)
43 \( 1 + (7.47 + 7.47i)T + 43iT^{2} \)
47 \( 1 + (-2.88 + 2.88i)T - 47iT^{2} \)
53 \( 1 + (2.46 + 2.46i)T + 53iT^{2} \)
59 \( 1 + 5.67T + 59T^{2} \)
61 \( 1 + (-1.16 + 1.16i)T - 61iT^{2} \)
67 \( 1 - 5.85T + 67T^{2} \)
71 \( 1 + (1.42 - 1.42i)T - 71iT^{2} \)
73 \( 1 + (-1.72 + 1.72i)T - 73iT^{2} \)
79 \( 1 + 7.43iT - 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + (-10.4 + 10.4i)T - 89iT^{2} \)
97 \( 1 + (0.456 - 0.456i)T - 97iT^{2} \)
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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

−10.59721213046908643452088295496, −9.414166668798989161578304840066, −8.387411979096223735693084690515, −7.71931260765807148287863282814, −7.41659295007004451060234998589, −5.45890452795709691854515305711, −4.71187807852485406212226389057, −3.32344824535158868005497160470, −2.21202786960111987339980093594, −0.45637318751531335795124329113, 1.15099754343077146384588443341, 3.62280329072854172802805339753, 4.74211335972188783596764758212, 5.14183431766645255695860820597, 6.85735963095935902143904390652, 7.59528510301477667524820007940, 7.893699612588472911749107525433, 9.053690161105739479860782620197, 10.08965911615503715011705272454, 10.93354393483640312564405483042

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