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Properties

Label 2-252-7.5-c8-0-24
Degree $2$
Conductor $252$
Sign $-0.922 - 0.385i$
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  + (−846. − 488. i)5-s + (−1.70e3 + 1.68e3i)7-s + (−8.63e3 − 1.49e4i)11-s − 4.54e4i·13-s + (1.12e5 − 6.48e4i)17-s + (1.36e5 + 7.90e4i)19-s + (1.76e5 − 3.05e5i)23-s + (2.82e5 + 4.89e5i)25-s − 2.56e5·29-s + (−1.52e5 + 8.82e4i)31-s + (2.27e6 − 5.91e5i)35-s + (1.07e6 − 1.85e6i)37-s − 4.55e6i·41-s − 5.94e6·43-s + (2.16e6 + 1.25e6i)47-s + ⋯
L(s)  = 1  + (−1.35 − 0.782i)5-s + (−0.711 + 0.702i)7-s + (−0.590 − 1.02i)11-s − 1.59i·13-s + (1.34 − 0.776i)17-s + (1.05 + 0.606i)19-s + (0.630 − 1.09i)23-s + (0.723 + 1.25i)25-s − 0.362·29-s + (−0.165 + 0.0955i)31-s + (1.51 − 0.394i)35-s + (0.571 − 0.989i)37-s − 1.61i·41-s − 1.73·43-s + (0.444 + 0.256i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.385i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.922 - 0.385i$
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: \((2,\ 252,\ (\ :4),\ -0.922 - 0.385i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.7270621064\)
\(L(\frac12)\) \(\approx\) \(0.7270621064\)
\(L(5)\) 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 \( 1 + (1.70e3 - 1.68e3i)T \)
good5 \( 1 + (846. + 488. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (8.63e3 + 1.49e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 4.54e4iT - 8.15e8T^{2} \)
17 \( 1 + (-1.12e5 + 6.48e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.36e5 - 7.90e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.76e5 + 3.05e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 2.56e5T + 5.00e11T^{2} \)
31 \( 1 + (1.52e5 - 8.82e4i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.07e6 + 1.85e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 4.55e6iT - 7.98e12T^{2} \)
43 \( 1 + 5.94e6T + 1.16e13T^{2} \)
47 \( 1 + (-2.16e6 - 1.25e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (8.11e5 + 1.40e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (9.69e6 - 5.59e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (3.54e6 + 2.04e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-5.10e5 - 8.84e5i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 9.23e6T + 6.45e14T^{2} \)
73 \( 1 + (-2.42e7 + 1.40e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-9.34e5 + 1.61e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 8.27e6iT - 2.25e15T^{2} \)
89 \( 1 + (1.18e7 + 6.84e6i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 9.99e7iT - 7.83e15T^{2} \)
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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

−10.08205440397840774689933180409, −8.914830144961840506343730054751, −8.102964688102623907054550324264, −7.43697859050203174144425497644, −5.73186267280209547296437453081, −5.14909492591482841415200389715, −3.51643621836793354352058432311, −2.97386310085112789130450357722, −0.78242123502789241017917321594, −0.23613230515611778599389030922, 1.33578131319554111590991491192, 3.04474143329404452286406829039, 3.78783285304842138942341088444, 4.84666856838006201376628148645, 6.54193274043544000436208785035, 7.30828464156974565721207090022, 7.86673471734023385338117805930, 9.454372046899714260751947275650, 10.15782983445961979896402280719, 11.31274181069970745458818051131

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