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Properties

Label 2-252-7.5-c8-0-21
Degree $2$
Conductor $252$
Sign $-0.790 + 0.612i$
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  + (−485. − 280. i)5-s + (−622. − 2.31e3i)7-s + (−5.55e3 − 9.62e3i)11-s + 3.86e4i·13-s + (9.18e4 − 5.30e4i)17-s + (1.64e5 + 9.47e4i)19-s + (2.31e5 − 4.01e5i)23-s + (−3.82e4 − 6.63e4i)25-s + 6.65e4·29-s + (4.71e5 − 2.72e5i)31-s + (−3.47e5 + 1.29e6i)35-s + (4.59e5 − 7.96e5i)37-s + 2.89e6i·41-s + 3.05e6·43-s + (9.60e4 + 5.54e4i)47-s + ⋯
L(s)  = 1  + (−0.776 − 0.448i)5-s + (−0.259 − 0.965i)7-s + (−0.379 − 0.657i)11-s + 1.35i·13-s + (1.09 − 0.635i)17-s + (1.25 + 0.727i)19-s + (0.828 − 1.43i)23-s + (−0.0980 − 0.169i)25-s + 0.0941·29-s + (0.510 − 0.294i)31-s + (−0.231 + 0.866i)35-s + (0.245 − 0.425i)37-s + 1.02i·41-s + 0.893·43-s + (0.0196 + 0.0113i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.790 + 0.612i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.266353531\)
\(L(\frac12)\) \(\approx\) \(1.266353531\)
\(L(5)\) 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 \( 1 + (622. + 2.31e3i)T \)
good5 \( 1 + (485. + 280. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (5.55e3 + 9.62e3i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 3.86e4iT - 8.15e8T^{2} \)
17 \( 1 + (-9.18e4 + 5.30e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.64e5 - 9.47e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-2.31e5 + 4.01e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 6.65e4T + 5.00e11T^{2} \)
31 \( 1 + (-4.71e5 + 2.72e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-4.59e5 + 7.96e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 2.89e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.05e6T + 1.16e13T^{2} \)
47 \( 1 + (-9.60e4 - 5.54e4i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (6.87e6 + 1.19e7i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (5.62e5 - 3.24e5i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (7.64e6 + 4.41e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-2.44e6 - 4.24e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 1.20e7T + 6.45e14T^{2} \)
73 \( 1 + (2.76e7 - 1.59e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-3.40e7 + 5.89e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 4.61e7iT - 2.25e15T^{2} \)
89 \( 1 + (9.23e6 + 5.33e6i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 4.77e7iT - 7.83e15T^{2} \)
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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

−10.21757172287922130638760092837, −9.306260328979859095900099455801, −8.138103969605756052271231728134, −7.42003478739418095141330354304, −6.33467931441626056765412698837, −4.93117665758899746879105965067, −4.01761851603141296334425092922, −2.98262418982011542417150797783, −1.17377851236573383110211905289, −0.33591237253999754095161220681, 1.13839679747803770812280738637, 2.80721838130490371464542365084, 3.43544580483658803652406745960, 5.07822091815624673039835969655, 5.83804715880292977648402659757, 7.36128932533827153070389431923, 7.83343796576088783123362502316, 9.120643375523949869446483318269, 10.03564106526501766457500040945, 11.03879446337385365687943265722

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