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Properties

Label 2-252-63.41-c3-0-10
Degree $2$
Conductor $252$
Sign $0.850 - 0.525i$
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  + (−4.09 + 3.19i)3-s + (3.53 − 6.12i)5-s + (10.8 + 15.0i)7-s + (6.56 − 26.1i)9-s + (7.40 − 4.27i)11-s + (−45.3 − 26.1i)13-s + (5.08 + 36.3i)15-s + 38.9·17-s − 66.4i·19-s + (−92.3 − 26.9i)21-s + (173. + 100. i)23-s + (37.5 + 64.9i)25-s + (56.7 + 128. i)27-s + (52.9 − 30.5i)29-s + (116. + 67.2i)31-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)3-s + (0.316 − 0.547i)5-s + (0.584 + 0.811i)7-s + (0.243 − 0.969i)9-s + (0.202 − 0.117i)11-s + (−0.967 − 0.558i)13-s + (0.0875 + 0.626i)15-s + 0.555·17-s − 0.801i·19-s + (−0.960 − 0.279i)21-s + (1.57 + 0.910i)23-s + (0.300 + 0.519i)25-s + (0.404 + 0.914i)27-s + (0.339 − 0.195i)29-s + (0.674 + 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.536638900\)
\(L(\frac12)\) \(\approx\) \(1.536638900\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.09 - 3.19i)T \)
7 \( 1 + (-10.8 - 15.0i)T \)
good5 \( 1 + (-3.53 + 6.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.40 + 4.27i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (45.3 + 26.1i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 38.9T + 4.91e3T^{2} \)
19 \( 1 + 66.4iT - 6.85e3T^{2} \)
23 \( 1 + (-173. - 100. i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-52.9 + 30.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-116. - 67.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 298.T + 5.06e4T^{2} \)
41 \( 1 + (221. - 383. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-26.1 - 45.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-137. - 238. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 136. iT - 1.48e5T^{2} \)
59 \( 1 + (-191. + 331. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-261. + 151. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-318. + 552. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 228. iT - 3.57e5T^{2} \)
73 \( 1 + 1.24e3iT - 3.89e5T^{2} \)
79 \( 1 + (100. + 174. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-323. - 560. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 826.T + 7.04e5T^{2} \)
97 \( 1 + (17.0 - 9.85i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.64118899671787912230021610269, −10.86534282093942696603184642209, −9.631271158195480874826186825490, −9.103624804072474491099691439549, −7.79895509029709657057981708552, −6.42083660393612286084148005922, −5.20223977264793482872530364442, −4.85213718537304541690294407272, −2.99370448691586732131775876661, −1.05099411599355147966145923169, 0.917018273976156710959818210691, 2.39591134802956191716542201695, 4.32183829599832020757716624915, 5.39177780908105469547109866245, 6.71761191319717559511636878202, 7.23028080241000692395083239021, 8.380957825441991439948027637263, 9.982005413814641097177499579936, 10.58129999744831829181462189315, 11.55070579912410030260599745584

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