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Properties

Label 2-252-28.27-c3-0-55
Degree $2$
Conductor $252$
Sign $-0.763 - 0.646i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
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  + (1.72 − 2.24i)2-s + (−2.05 − 7.73i)4-s − 6.58i·5-s + (−15.1 + 10.5i)7-s + (−20.8 − 8.73i)8-s + (−14.7 − 11.3i)10-s − 54.2i·11-s + 40.9i·13-s + (−2.45 + 52.3i)14-s + (−55.5 + 31.7i)16-s + 69.4i·17-s − 160.·19-s + (−50.9 + 13.5i)20-s + (−121. − 93.5i)22-s − 87.8i·23-s + ⋯
L(s)  = 1  + (0.609 − 0.792i)2-s + (−0.256 − 0.966i)4-s − 0.589i·5-s + (−0.820 + 0.571i)7-s + (−0.922 − 0.386i)8-s + (−0.467 − 0.359i)10-s − 1.48i·11-s + 0.874i·13-s + (−0.0468 + 0.998i)14-s + (−0.868 + 0.495i)16-s + 0.991i·17-s − 1.93·19-s + (−0.569 + 0.151i)20-s + (−1.17 − 0.906i)22-s − 0.796i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.646i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.763 - 0.646i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.763 - 0.646i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7937271423\)
\(L(\frac12)\) \(\approx\) \(0.7937271423\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.72 + 2.24i)T \)
3 \( 1 \)
7 \( 1 + (15.1 - 10.5i)T \)
good5 \( 1 + 6.58iT - 125T^{2} \)
11 \( 1 + 54.2iT - 1.33e3T^{2} \)
13 \( 1 - 40.9iT - 2.19e3T^{2} \)
17 \( 1 - 69.4iT - 4.91e3T^{2} \)
19 \( 1 + 160.T + 6.85e3T^{2} \)
23 \( 1 + 87.8iT - 1.21e4T^{2} \)
29 \( 1 + 236.T + 2.43e4T^{2} \)
31 \( 1 - 131.T + 2.97e4T^{2} \)
37 \( 1 + 23.6T + 5.06e4T^{2} \)
41 \( 1 + 112. iT - 6.89e4T^{2} \)
43 \( 1 + 194. iT - 7.95e4T^{2} \)
47 \( 1 + 269.T + 1.03e5T^{2} \)
53 \( 1 - 120.T + 1.48e5T^{2} \)
59 \( 1 + 338.T + 2.05e5T^{2} \)
61 \( 1 - 267. iT - 2.26e5T^{2} \)
67 \( 1 - 275. iT - 3.00e5T^{2} \)
71 \( 1 + 270. iT - 3.57e5T^{2} \)
73 \( 1 + 1.23e3iT - 3.89e5T^{2} \)
79 \( 1 + 691. iT - 4.93e5T^{2} \)
83 \( 1 - 430.T + 5.71e5T^{2} \)
89 \( 1 + 1.22e3iT - 7.04e5T^{2} \)
97 \( 1 + 381. iT - 9.12e5T^{2} \)
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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

−11.07664959378723193805782327445, −10.31664404079430252210690045699, −8.948695168447553856111781466551, −8.652281980052032864477021138760, −6.45279972373999177503515370263, −5.87796792970910854915153496831, −4.48508739959384622958519894183, −3.40765294945440282483746997631, −2.00626721630975464769604351103, −0.23959872804400547814933205489, 2.63593125298196362102073227505, 3.86679895494704901617123944822, 4.98382358802694239788532092061, 6.36760234356821567516878919398, 7.05473193318077811513546434300, 7.88271284966532416119602700794, 9.303413669582937202913103986155, 10.18980643404295711096730427621, 11.32175811203194567447241868720, 12.68630226195272132847976976614

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