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Properties

Label 2-624-13.3-c1-0-8
Degree $2$
Conductor $624$
Sign $0.128 + 0.991i$
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 − 3.56·5-s + (0.280 − 0.486i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + (0.5 − 3.57i)13-s + (−1.78 − 3.08i)15-s + (0.780 − 1.35i)17-s + (3.56 − 6.16i)19-s + 0.561·21-s + (1 + 1.73i)23-s + 7.68·25-s − 0.999·27-s + (−3.34 − 5.78i)29-s − 2.56·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 1.59·5-s + (0.106 − 0.183i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + (0.138 − 0.990i)13-s + (−0.459 − 0.796i)15-s + (0.189 − 0.327i)17-s + (0.817 − 1.41i)19-s + 0.122·21-s + (0.208 + 0.361i)23-s + 1.53·25-s − 0.192·27-s + (−0.620 − 1.07i)29-s − 0.460·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.128 + 0.991i$
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: \((2,\ 624,\ (\ :1/2),\ 0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630134 - 0.553624i\)
\(L(\frac12)\) \(\approx\) \(0.630134 - 0.553624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 3.57i)T \)
good5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 + (-0.280 + 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.780 + 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 + 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 + 5.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + (3.78 + 6.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.780 - 1.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.28 + 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 + 0.684T + 53T^{2} \)
59 \( 1 + (1.43 - 2.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.28 - 3.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 5.43T + 79T^{2} \)
83 \( 1 - 0.876T + 83T^{2} \)
89 \( 1 + (2.43 + 4.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.28 + 7.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.62566963731785661378625907271, −9.443878905407406786139334590508, −8.598398770522860092387520706709, −7.71925369265270614890965488318, −7.26771426300015864789052285630, −5.64899908922128161624955563255, −4.68669119058583451964231975684, −3.67022775193803398915335130199, −2.91565461002953126131692825179, −0.46050964329215285909052653477, 1.60596056140616494126116877325, 3.25118976497080230390722571354, 4.06979886542942372127697709259, 5.21612815132155353422920144869, 6.61635712804367118797033260190, 7.42014728539716795639746611385, 8.069550674286935861676031394230, 8.819093028947583845615768355081, 9.913397301458806992003294109777, 11.02928908191205760133702895566

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