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Properties

Label 2-624-39.32-c1-0-14
Degree $2$
Conductor $624$
Sign $0.884 - 0.466i$
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  + (1.43 + 0.963i)3-s + (0.504 + 0.504i)5-s + (0.836 − 3.12i)7-s + (1.14 + 2.77i)9-s + (0.296 + 1.10i)11-s + (3.23 + 1.58i)13-s + (0.239 + 1.21i)15-s + (1.51 + 2.63i)17-s + (−4.59 − 1.23i)19-s + (4.21 − 3.68i)21-s + (2.43 − 4.21i)23-s − 4.49i·25-s + (−1.03 + 5.09i)27-s + (8.98 + 5.18i)29-s + (1.93 − 1.93i)31-s + ⋯
L(s)  = 1  + (0.830 + 0.556i)3-s + (0.225 + 0.225i)5-s + (0.316 − 1.18i)7-s + (0.380 + 0.924i)9-s + (0.0894 + 0.333i)11-s + (0.897 + 0.440i)13-s + (0.0618 + 0.313i)15-s + (0.368 + 0.638i)17-s + (−1.05 − 0.282i)19-s + (0.919 − 0.804i)21-s + (0.506 − 0.877i)23-s − 0.898i·25-s + (−0.198 + 0.980i)27-s + (1.66 + 0.962i)29-s + (0.346 − 0.346i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.884 - 0.466i$
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: \((2,\ 624,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10011 + 0.520445i\)
\(L(\frac12)\) \(\approx\) \(2.10011 + 0.520445i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 - 0.963i)T \)
13 \( 1 + (-3.23 - 1.58i)T \)
good5 \( 1 + (-0.504 - 0.504i)T + 5iT^{2} \)
7 \( 1 + (-0.836 + 3.12i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.296 - 1.10i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.51 - 2.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.59 + 1.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.43 + 4.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.98 - 5.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.93 + 1.93i)T - 31iT^{2} \)
37 \( 1 + (7.49 - 2.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.55 - 1.75i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.98 - 1.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.27 - 1.27i)T - 47iT^{2} \)
53 \( 1 - 2.42iT - 53T^{2} \)
59 \( 1 + (-5.61 - 1.50i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.23 + 9.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.57 - 5.88i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.02 - 3.83i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.06 + 4.06i)T + 73iT^{2} \)
79 \( 1 + 6.77T + 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 + (2.07 + 7.73i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.989 + 0.265i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.32838182164587849204747937835, −10.17890426314192244580547567813, −8.668680748199526470200148212599, −8.383553952793753235684669806319, −7.12090617786479857862758266322, −6.39582468877433787617919518401, −4.73782215901253558195567555539, −4.14391937098457953313194989854, −3.01664589299332590352333424221, −1.57916937711857734809745987154, 1.40450522485583610445988448098, 2.61853492956672981884748159653, 3.63195018636265555089448965237, 5.15623905129725482049758301922, 6.04769270741838847620196729206, 7.01588949529458890616619066953, 8.278313667758914464551050535947, 8.583051569031981347375079150565, 9.416908467759510644002746593223, 10.44122506054742608394003719649

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