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Properties

Label 2-624-13.3-c1-0-7
Degree $2$
Conductor $624$
Sign $0.872 - 0.488i$
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  + (0.5 + 0.866i)3-s + 3·5-s + (1 − 1.73i)7-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)11-s + (−3.5 + 0.866i)13-s + (1.5 + 2.59i)15-s + (1.5 − 2.59i)17-s + (1 − 1.73i)19-s + 1.99·21-s + (−3 − 5.19i)23-s + 4·25-s − 0.999·27-s + (−1.5 − 2.59i)29-s + 4·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 1.34·5-s + (0.377 − 0.654i)7-s + (−0.166 + 0.288i)9-s + (0.904 + 1.56i)11-s + (−0.970 + 0.240i)13-s + (0.387 + 0.670i)15-s + (0.363 − 0.630i)17-s + (0.229 − 0.397i)19-s + 0.436·21-s + (−0.625 − 1.08i)23-s + 0.800·25-s − 0.192·27-s + (−0.278 − 0.482i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04469 + 0.533847i\)
\(L(\frac12)\) \(\approx\) \(2.04469 + 0.533847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (3.5 - 0.866i)T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.28640645628495921856398236351, −9.716887311965313403743232045366, −9.424154880918457675126148905083, −8.050007403862835446128800774610, −7.06523591181107522302721005016, −6.24720655627968707306856005175, −4.84463068511516099949861750980, −4.43442685339367085143955114496, −2.71126122125106236736270124767, −1.65603438242717813851675669160, 1.41009298425134880342500189697, 2.46571490286286266500513606038, 3.69194797962191056901510713480, 5.56980462981863390959685445144, 5.74553235032811395336553220431, 6.87411254106722560239472779653, 8.058424623744113190099056541143, 8.823672827469495560122281666167, 9.568997773695613993861017676474, 10.39630604373407081910258869264

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