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Properties

Label 2-624-156.83-c0-0-0
Degree $2$
Conductor $624$
Sign $-0.289 - 0.957i$
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  + i·3-s + (−1 + i)7-s − 9-s + i·13-s + (1 + i)19-s + (−1 − i)21-s − i·25-s − i·27-s + (−1 − i)31-s + (1 + i)37-s − 39-s + 2·43-s − i·49-s + (−1 + i)57-s + (1 − i)63-s + ⋯
L(s)  = 1  + i·3-s + (−1 + i)7-s − 9-s + i·13-s + (1 + i)19-s + (−1 − i)21-s − i·25-s − i·27-s + (−1 − i)31-s + (1 + i)37-s − 39-s + 2·43-s − i·49-s + (−1 + i)57-s + (1 − i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7744806690\)
\(L(\frac12)\) \(\approx\) \(0.7744806690\)
\(L(1)\) 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 \( 1 - iT \)
13 \( 1 - iT \)
good5 \( 1 + iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−11.05166707889653930175786089572, −9.944540973359718320332965974933, −9.459302909875637121571843867806, −8.790782976908530853575516345118, −7.67518730519710932554227260605, −6.25841457990201291275324435694, −5.74343930676503148968667070993, −4.51227218613201695542567394735, −3.50819527704168511020135109670, −2.43561245836311904086621093138, 0.919220630484842087631974528060, 2.74767561213197130896813141058, 3.66250277149528561884433183962, 5.26447252812542490775105176132, 6.17099332245191928173702122491, 7.35591247031106411132760338693, 7.41116142691185022162072803189, 8.841630836075293556161477061963, 9.631990022786358968870835853678, 10.71197930365671610953105604499

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