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Properties

Label 2-624-39.32-c1-0-17
Degree $2$
Conductor $624$
Sign $0.967 - 0.254i$
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  + (1.73 + 0.0795i)3-s + (2.76 + 2.76i)5-s + (0.657 − 2.45i)7-s + (2.98 + 0.275i)9-s + (−0.150 − 0.563i)11-s + (−1.20 − 3.39i)13-s + (4.56 + 5.00i)15-s + (−0.547 − 0.947i)17-s + (−1.32 − 0.355i)19-s + (1.33 − 4.19i)21-s + (−0.876 + 1.51i)23-s + 10.2i·25-s + (5.14 + 0.713i)27-s + (−5.12 − 2.96i)29-s + (−6.49 + 6.49i)31-s + ⋯
L(s)  = 1  + (0.998 + 0.0459i)3-s + (1.23 + 1.23i)5-s + (0.248 − 0.927i)7-s + (0.995 + 0.0917i)9-s + (−0.0454 − 0.169i)11-s + (−0.335 − 0.942i)13-s + (1.17 + 1.29i)15-s + (−0.132 − 0.229i)17-s + (−0.304 − 0.0815i)19-s + (0.290 − 0.915i)21-s + (−0.182 + 0.316i)23-s + 2.05i·25-s + (0.990 + 0.137i)27-s + (−0.952 − 0.549i)29-s + (−1.16 + 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.967 - 0.254i$
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: \((2,\ 624,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50230 + 0.323654i\)
\(L(\frac12)\) \(\approx\) \(2.50230 + 0.323654i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0795i)T \)
13 \( 1 + (1.20 + 3.39i)T \)
good5 \( 1 + (-2.76 - 2.76i)T + 5iT^{2} \)
7 \( 1 + (-0.657 + 2.45i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.150 + 0.563i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.547 + 0.947i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 + 0.355i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.876 - 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.12 + 2.96i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.49 - 6.49i)T - 31iT^{2} \)
37 \( 1 + (2.98 - 0.801i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.11 + 1.37i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.26 - 1.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.51 - 5.51i)T - 47iT^{2} \)
53 \( 1 - 3.04iT - 53T^{2} \)
59 \( 1 + (-8.19 - 2.19i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.67 + 8.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.70 + 6.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.220 + 0.821i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.18 + 5.18i)T + 73iT^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + (5.15 + 5.15i)T + 83iT^{2} \)
89 \( 1 + (-2.50 - 9.35i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.592 + 0.158i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.53948233111958651349165120166, −9.837033920655686072944153174806, −9.100951240209083513097346871920, −7.81202824215798045700153828389, −7.23755585316806981767013089818, −6.31779617985782625109409035365, −5.14238694992501803523411815372, −3.70348299319523553248336107995, −2.83637710787264024925222931084, −1.74630958802606834985780791345, 1.74922901957988657419277681386, 2.28714846226626619920987247336, 4.03237364046791891473496691738, 5.05839102162815189334132909516, 5.90147555077211313986202381922, 7.08549613367751413787900996932, 8.317403983005872485722802087802, 8.941226518215236166392324837113, 9.410868557507385341700735008572, 10.18319279212192888650637570010

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