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Properties

Label 2-252-7.2-c11-0-23
Degree $2$
Conductor $252$
Sign $0.483 + 0.875i$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.41e3 + 7.64e3i)5-s + (−3.65e4 − 2.53e4i)7-s + (1.63e5 + 2.83e5i)11-s + 1.62e6·13-s + (−1.31e6 − 2.26e6i)17-s + (3.70e6 − 6.41e6i)19-s + (5.76e6 − 9.98e6i)23-s + (−1.45e7 − 2.52e7i)25-s − 1.97e8·29-s + (1.08e8 + 1.88e8i)31-s + (3.55e8 − 1.67e8i)35-s + (−3.38e8 + 5.85e8i)37-s − 1.07e9·41-s + 3.16e8·43-s + (−1.43e9 + 2.49e9i)47-s + ⋯
L(s)  = 1  + (−0.631 + 1.09i)5-s + (−0.822 − 0.569i)7-s + (0.306 + 0.530i)11-s + 1.21·13-s + (−0.223 − 0.387i)17-s + (0.342 − 0.594i)19-s + (0.186 − 0.323i)23-s + (−0.298 − 0.517i)25-s − 1.79·29-s + (0.682 + 1.18i)31-s + (1.14 − 0.540i)35-s + (−0.801 + 1.38i)37-s − 1.44·41-s + 0.328·43-s + (−0.915 + 1.58i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.8615368386\)
\(L(\frac12)\) \(\approx\) \(0.8615368386\)
\(L(\frac{13}{2})\) 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 \( 1 \)
7 \( 1 + (3.65e4 + 2.53e4i)T \)
good5 \( 1 + (4.41e3 - 7.64e3i)T + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (-1.63e5 - 2.83e5i)T + (-1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 1.62e6T + 1.79e12T^{2} \)
17 \( 1 + (1.31e6 + 2.26e6i)T + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (-3.70e6 + 6.41e6i)T + (-5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-5.76e6 + 9.98e6i)T + (-4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 1.97e8T + 1.22e16T^{2} \)
31 \( 1 + (-1.08e8 - 1.88e8i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (3.38e8 - 5.85e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 + 1.07e9T + 5.50e17T^{2} \)
43 \( 1 - 3.16e8T + 9.29e17T^{2} \)
47 \( 1 + (1.43e9 - 2.49e9i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (6.06e8 + 1.05e9i)T + (-4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (5.29e9 + 9.16e9i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (-5.87e9 + 1.01e10i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (3.68e8 + 6.38e8i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 + 1.60e10T + 2.31e20T^{2} \)
73 \( 1 + (5.44e9 + 9.43e9i)T + (-1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (1.18e10 - 2.04e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 - 5.25e10T + 1.28e21T^{2} \)
89 \( 1 + (8.77e8 - 1.51e9i)T + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 - 7.32e10T + 7.15e21T^{2} \)
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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

−10.05224770092039125732195480196, −9.073877389748864490919715206529, −7.84240049233301570939435231721, −6.84640531053221433152452698792, −6.44727041039023491364569267833, −4.81889769874945144518209571125, −3.54579247282942564771716693978, −3.12581142663277741843363931462, −1.55131376920736651563699244436, −0.21983533427045785290388746909, 0.72034366851833411200426337489, 1.81767576974157291990962882493, 3.41726740311899153641523738495, 4.03663464918262216472406794248, 5.46840412632800609816579522670, 6.13933553184678923319355768748, 7.49008091259763670533779855176, 8.663427292813160466650257276115, 8.961517414745225790617510229054, 10.19195373971185885531331877199

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