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Properties

Label 2-624-39.32-c1-0-19
Degree $2$
Conductor $624$
Sign $-0.814 + 0.580i$
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  + (−1.55 − 0.764i)3-s + (−0.504 − 0.504i)5-s + (0.836 − 3.12i)7-s + (1.83 + 2.37i)9-s + (−0.296 − 1.10i)11-s + (3.23 + 1.58i)13-s + (0.398 + 1.16i)15-s + (−1.51 − 2.63i)17-s + (−4.59 − 1.23i)19-s + (−3.68 + 4.21i)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  + (−0.897 − 0.441i)3-s + (−0.225 − 0.225i)5-s + (0.316 − 1.18i)7-s + (0.610 + 0.791i)9-s + (−0.0894 − 0.333i)11-s + (0.897 + 0.440i)13-s + (0.102 + 0.302i)15-s + (−0.368 − 0.638i)17-s + (−1.05 − 0.282i)19-s + (−0.804 + 0.919i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.814 + 0.580i$
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: \((2,\ 624,\ (\ :1/2),\ -0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217913 - 0.681355i\)
\(L(\frac12)\) \(\approx\) \(0.217913 - 0.681355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 0.764i)T \)
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

−10.58568604430704745426080081548, −9.500861498170360138917378213055, −8.290057826545023772132569537354, −7.52876983374656435389020995226, −6.66084197922442770210839816910, −5.81688613194321643159028214211, −4.60012829999950817027744189261, −3.87791452157685519402988102674, −1.86492791390306189749942766683, −0.43808627986276283257029017442, 1.84349534895702285991502802333, 3.48214283960160632330630749402, 4.54577615720457212735466038329, 5.64272637839775235879090999122, 6.17506650333709547928034391806, 7.32134403155718233546726422241, 8.566212737885460991302817980353, 9.121205046191135382409592312079, 10.42569132599243400791779409604, 10.86049271971246448197881913744

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