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Properties

Label 2-624-39.11-c1-0-18
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: $\chi_{624} (401, \cdot )$
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

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

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