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Properties

Label 2-252-21.11-c8-0-18
Degree $2$
Conductor $252$
Sign $-0.234 + 0.972i$
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  + (724. − 418. i)5-s + (613. + 2.32e3i)7-s + (1.82e4 + 1.05e4i)11-s − 4.89e4·13-s + (−5.72e4 − 3.30e4i)17-s + (−1.24e5 − 2.16e5i)19-s + (−3.18e4 + 1.84e4i)23-s + (1.54e5 − 2.68e5i)25-s − 1.59e5i·29-s + (4.58e5 − 7.93e5i)31-s + (1.41e6 + 1.42e6i)35-s + (−6.70e5 − 1.16e6i)37-s − 1.76e6i·41-s − 2.70e6·43-s + (7.31e6 − 4.22e6i)47-s + ⋯
L(s)  = 1  + (1.15 − 0.669i)5-s + (0.255 + 0.966i)7-s + (1.24 + 0.720i)11-s − 1.71·13-s + (−0.685 − 0.395i)17-s + (−0.957 − 1.65i)19-s + (−0.113 + 0.0657i)23-s + (0.396 − 0.686i)25-s − 0.226i·29-s + (0.496 − 0.859i)31-s + (0.943 + 0.950i)35-s + (−0.357 − 0.619i)37-s − 0.623i·41-s − 0.790·43-s + (1.50 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.234 + 0.972i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.234 + 0.972i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.789411067\)
\(L(\frac12)\) \(\approx\) \(1.789411067\)
\(L(5)\) 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 \( 1 + (-613. - 2.32e3i)T \)
good5 \( 1 + (-724. + 418. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.82e4 - 1.05e4i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 4.89e4T + 8.15e8T^{2} \)
17 \( 1 + (5.72e4 + 3.30e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.24e5 + 2.16e5i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (3.18e4 - 1.84e4i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.59e5iT - 5.00e11T^{2} \)
31 \( 1 + (-4.58e5 + 7.93e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (6.70e5 + 1.16e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 1.76e6iT - 7.98e12T^{2} \)
43 \( 1 + 2.70e6T + 1.16e13T^{2} \)
47 \( 1 + (-7.31e6 + 4.22e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (4.99e6 + 2.88e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (4.25e6 + 2.45e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-4.36e6 - 7.56e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.27e6 - 2.20e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 8.62e6iT - 6.45e14T^{2} \)
73 \( 1 + (-2.18e6 + 3.79e6i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-1.66e7 - 2.89e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + 3.92e7iT - 2.25e15T^{2} \)
89 \( 1 + (-9.08e7 + 5.24e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 - 1.61e8T + 7.83e15T^{2} \)
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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

−10.04615114961178685600337400294, −9.226788744558425266337947341368, −8.861083235702514831892419745752, −7.23266682736538177377608990085, −6.29605979302006077514852560530, −5.14008129514430501818784594355, −4.48823204458410827699228998707, −2.39730605960519084601933365683, −1.95609975452674827048815691243, −0.35219998002048288561296570233, 1.26748107393765753239386035911, 2.23999560973841402407915047482, 3.59954997713249461195221951399, 4.74418024878712313294572712883, 6.16706288976065871559932881738, 6.71034346683850695354637324625, 7.900483740112353770077028112512, 9.128423199318148504459004410888, 10.15101303816626514200022313353, 10.56486791259659455350493952474

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