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Properties

Label 2-624-12.11-c3-0-0
Degree $2$
Conductor $624$
Sign $-0.546 + 0.837i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.35 + 4.63i)3-s − 6.29i·5-s + 30.3i·7-s + (−15.9 + 21.7i)9-s − 49.7·11-s − 13·13-s + (29.1 − 14.8i)15-s − 131. i·17-s + 115. i·19-s + (−140. + 71.3i)21-s − 74.5·23-s + 85.3·25-s + (−138. − 22.7i)27-s − 229. i·29-s − 67.7i·31-s + ⋯
L(s)  = 1  + (0.452 + 0.891i)3-s − 0.563i·5-s + 1.63i·7-s + (−0.590 + 0.806i)9-s − 1.36·11-s − 0.277·13-s + (0.502 − 0.254i)15-s − 1.87i·17-s + 1.39i·19-s + (−1.46 + 0.741i)21-s − 0.675·23-s + 0.682·25-s + (−0.986 − 0.162i)27-s − 1.46i·29-s − 0.392i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.546 + 0.837i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1356042758\)
\(L(\frac12)\) \(\approx\) \(0.1356042758\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.35 - 4.63i)T \)
13 \( 1 + 13T \)
good5 \( 1 + 6.29iT - 125T^{2} \)
7 \( 1 - 30.3iT - 343T^{2} \)
11 \( 1 + 49.7T + 1.33e3T^{2} \)
17 \( 1 + 131. iT - 4.91e3T^{2} \)
19 \( 1 - 115. iT - 6.85e3T^{2} \)
23 \( 1 + 74.5T + 1.21e4T^{2} \)
29 \( 1 + 229. iT - 2.43e4T^{2} \)
31 \( 1 + 67.7iT - 2.97e4T^{2} \)
37 \( 1 - 67.2T + 5.06e4T^{2} \)
41 \( 1 + 88.1iT - 6.89e4T^{2} \)
43 \( 1 + 334. iT - 7.95e4T^{2} \)
47 \( 1 + 97.2T + 1.03e5T^{2} \)
53 \( 1 - 684. iT - 1.48e5T^{2} \)
59 \( 1 + 33.3T + 2.05e5T^{2} \)
61 \( 1 + 305.T + 2.26e5T^{2} \)
67 \( 1 - 659. iT - 3.00e5T^{2} \)
71 \( 1 + 497.T + 3.57e5T^{2} \)
73 \( 1 - 131.T + 3.89e5T^{2} \)
79 \( 1 + 924. iT - 4.93e5T^{2} \)
83 \( 1 - 597.T + 5.71e5T^{2} \)
89 \( 1 - 166. iT - 7.04e5T^{2} \)
97 \( 1 + 1.45e3T + 9.12e5T^{2} \)
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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

−10.58064676173147760831755764876, −9.728038751787167786794685401719, −9.135434188035967595348074360565, −8.293846457738395215853325231515, −7.62942503744554274688835414239, −5.79906525622294712399623722250, −5.32900998544260732349039950486, −4.42411488679396616219554136841, −2.90713951032101247507567757889, −2.27923050779042968372849099207, 0.03507111346337799927359511853, 1.38917745897649652290910484586, 2.73306040023237307825061137721, 3.67618023793696717613324653325, 4.94564580418626240755083477437, 6.39555598448944878199483367059, 7.01114256305615606416957268574, 7.79253738030587073125014231991, 8.435662204463784061212366191663, 9.760997380062141158756847585342

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