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Properties

Label 2-252-7.3-c8-0-4
Degree $2$
Conductor $252$
Sign $-0.989 - 0.144i$
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  + (225. − 129. i)5-s + (597. + 2.32e3i)7-s + (−9.72e3 + 1.68e4i)11-s + 4.63e4i·13-s + (1.05e5 + 6.11e4i)17-s + (1.84e4 − 1.06e4i)19-s + (−1.52e5 − 2.63e5i)23-s + (−1.61e5 + 2.79e5i)25-s − 8.81e4·29-s + (−1.54e6 − 8.94e5i)31-s + (4.36e5 + 4.45e5i)35-s + (7.55e5 + 1.30e6i)37-s − 1.88e5i·41-s − 1.23e5·43-s + (5.58e6 − 3.22e6i)47-s + ⋯
L(s)  = 1  + (0.360 − 0.207i)5-s + (0.248 + 0.968i)7-s + (−0.664 + 1.15i)11-s + 1.62i·13-s + (1.26 + 0.731i)17-s + (0.141 − 0.0817i)19-s + (−0.543 − 0.942i)23-s + (−0.413 + 0.716i)25-s − 0.124·29-s + (−1.67 − 0.968i)31-s + (0.290 + 0.297i)35-s + (0.402 + 0.697i)37-s − 0.0665i·41-s − 0.0360·43-s + (1.14 − 0.661i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.989 - 0.144i$
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: \((2,\ 252,\ (\ :4),\ -0.989 - 0.144i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.227908067\)
\(L(\frac12)\) \(\approx\) \(1.227908067\)
\(L(5)\) 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 \( 1 + (-597. - 2.32e3i)T \)
good5 \( 1 + (-225. + 129. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (9.72e3 - 1.68e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 4.63e4iT - 8.15e8T^{2} \)
17 \( 1 + (-1.05e5 - 6.11e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-1.84e4 + 1.06e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (1.52e5 + 2.63e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 8.81e4T + 5.00e11T^{2} \)
31 \( 1 + (1.54e6 + 8.94e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-7.55e5 - 1.30e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 1.88e5iT - 7.98e12T^{2} \)
43 \( 1 + 1.23e5T + 1.16e13T^{2} \)
47 \( 1 + (-5.58e6 + 3.22e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-6.88e6 + 1.19e7i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (1.00e7 + 5.79e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (1.92e7 - 1.11e7i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (-5.75e6 + 9.97e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 3.11e7T + 6.45e14T^{2} \)
73 \( 1 + (1.08e7 + 6.26e6i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-2.13e7 - 3.69e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 7.19e7iT - 2.25e15T^{2} \)
89 \( 1 + (4.37e7 - 2.52e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 1.16e8iT - 7.83e15T^{2} \)
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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

−11.13716411494268382504755300909, −9.891306383773769573758514324579, −9.288432558318814131394679510895, −8.206474054435126018892941020719, −7.17508572014662435722130776303, −5.95413607858574998793743084859, −5.08785247486932536633457827077, −3.93861901917388585635975292866, −2.28988854546036275527212403355, −1.65157842255586435948164413370, 0.26247816492094654381160481443, 1.19119324440511837976364113882, 2.84915477327249426279959878758, 3.68483587432464587846439366793, 5.29895479799309588904698072260, 5.89570183443549026813901310240, 7.53599772607128253368950621936, 7.87300606833713825665604130855, 9.297105304348891152314969721146, 10.46388094530405121646756827372

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