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Properties

Label 2-252-7.3-c8-0-2
Degree $2$
Conductor $252$
Sign $-0.867 + 0.497i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−656. + 379. i)5-s + (1.90e3 + 1.45e3i)7-s + (−6.90e3 + 1.19e4i)11-s + 3.99e4i·13-s + (−3.30e4 − 1.91e4i)17-s + (−1.19e5 + 6.87e4i)19-s + (7.45e4 + 1.29e5i)23-s + (9.23e4 − 1.59e5i)25-s − 2.77e5·29-s + (1.48e6 + 8.59e5i)31-s + (−1.80e6 − 2.35e5i)35-s + (2.35e5 + 4.07e5i)37-s + 2.24e6i·41-s + 4.59e6·43-s + (−3.14e6 + 1.81e6i)47-s + ⋯
L(s)  = 1  + (−1.05 + 0.606i)5-s + (0.794 + 0.607i)7-s + (−0.471 + 0.817i)11-s + 1.39i·13-s + (−0.396 − 0.228i)17-s + (−0.913 + 0.527i)19-s + (0.266 + 0.461i)23-s + (0.236 − 0.409i)25-s − 0.392·29-s + (1.61 + 0.930i)31-s + (−1.20 − 0.156i)35-s + (0.125 + 0.217i)37-s + 0.796i·41-s + 1.34·43-s + (−0.644 + 0.372i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.867 + 0.497i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.867 + 0.497i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.9389146601\)
\(L(\frac12)\) \(\approx\) \(0.9389146601\)
\(L(5)\) 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 \( 1 \)
7 \( 1 + (-1.90e3 - 1.45e3i)T \)
good5 \( 1 + (656. - 379. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (6.90e3 - 1.19e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 3.99e4iT - 8.15e8T^{2} \)
17 \( 1 + (3.30e4 + 1.91e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.19e5 - 6.87e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (-7.45e4 - 1.29e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 2.77e5T + 5.00e11T^{2} \)
31 \( 1 + (-1.48e6 - 8.59e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-2.35e5 - 4.07e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 2.24e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.59e6T + 1.16e13T^{2} \)
47 \( 1 + (3.14e6 - 1.81e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-2.54e6 + 4.40e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (2.58e6 + 1.49e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-1.11e7 + 6.43e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (1.27e5 - 2.21e5i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 4.77e7T + 6.45e14T^{2} \)
73 \( 1 + (1.52e7 + 8.81e6i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (1.56e7 + 2.71e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 7.76e7iT - 2.25e15T^{2} \)
89 \( 1 + (-7.90e7 + 4.56e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 2.05e7iT - 7.83e15T^{2} \)
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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

−11.39819601974085022911648081030, −10.35839984677422269232400832627, −9.141434598992391374959474356639, −8.169890097093741419166428526675, −7.33370784304415762769020861909, −6.37963987924388059489419472401, −4.85284527287536291000546829423, −4.11339306182889712515731217521, −2.68043971785489176216094337473, −1.60984060457892030224392673372, 0.25694017352100454447686108361, 0.867703998631653654227027019435, 2.61276176915678032728375033338, 3.95979619296211288824034818459, 4.74104934655128902187177089122, 5.90917409845394007147052945775, 7.39511959075701690733601735479, 8.163547905182751159529008772613, 8.697323049008643162891698695396, 10.35409086744049904432140892692

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