Properties

Label 2-252-7.4-c11-0-1
Degree $2$
Conductor $252$
Sign $0.252 - 0.967i$
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  + (−4.94e3 − 8.57e3i)5-s + (2.91e4 − 3.35e4i)7-s + (−2.50e4 + 4.33e4i)11-s + 1.46e6·13-s + (−3.16e6 + 5.48e6i)17-s + (−1.02e7 − 1.77e7i)19-s + (2.36e7 + 4.08e7i)23-s + (−2.45e7 + 4.25e7i)25-s + 9.10e7·29-s + (−4.83e7 + 8.37e7i)31-s + (−4.32e8 − 8.38e7i)35-s + (−7.14e7 − 1.23e8i)37-s − 8.44e8·41-s − 1.55e9·43-s + (4.22e8 + 7.32e8i)47-s + ⋯
L(s)  = 1  + (−0.708 − 1.22i)5-s + (0.655 − 0.754i)7-s + (−0.0468 + 0.0810i)11-s + 1.09·13-s + (−0.541 + 0.937i)17-s + (−0.949 − 1.64i)19-s + (0.764 + 1.32i)23-s + (−0.503 + 0.872i)25-s + 0.824·29-s + (−0.303 + 0.525i)31-s + (−1.39 − 0.269i)35-s + (−0.169 − 0.293i)37-s − 1.13·41-s − 1.61·43-s + (0.268 + 0.465i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.6611598767\)
\(L(\frac12)\) \(\approx\) \(0.6611598767\)
\(L(\frac{13}{2})\) 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 \( 1 + (-2.91e4 + 3.35e4i)T \)
good5 \( 1 + (4.94e3 + 8.57e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (2.50e4 - 4.33e4i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 - 1.46e6T + 1.79e12T^{2} \)
17 \( 1 + (3.16e6 - 5.48e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (1.02e7 + 1.77e7i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (-2.36e7 - 4.08e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 - 9.10e7T + 1.22e16T^{2} \)
31 \( 1 + (4.83e7 - 8.37e7i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (7.14e7 + 1.23e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 8.44e8T + 5.50e17T^{2} \)
43 \( 1 + 1.55e9T + 9.29e17T^{2} \)
47 \( 1 + (-4.22e8 - 7.32e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (1.26e9 - 2.19e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (2.80e9 - 4.85e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (5.98e8 + 1.03e9i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (2.73e9 - 4.73e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 1.81e10T + 2.31e20T^{2} \)
73 \( 1 + (-9.87e9 + 1.71e10i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (2.14e10 + 3.71e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 5.21e10T + 1.28e21T^{2} \)
89 \( 1 + (-1.39e10 - 2.42e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 5.42e10T + 7.15e21T^{2} \)
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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

−10.53596680793216918806658899174, −8.942372588573743976854934965038, −8.563337354451433001724532630629, −7.55664302142031352611697421868, −6.47303476123965202259041656054, −5.01460801897747808070023615686, −4.42393328894407981803585458599, −3.44127690376537716376727078651, −1.66702598156921249451612066197, −0.929671569547179210966137904742, 0.13618873718443771924240044316, 1.68973818242346126199257361248, 2.76479299310505325712293193520, 3.69570189225341219196525301400, 4.85106682430001775970166289576, 6.17387918425139620246665544931, 6.88233785209363598999835157064, 8.144468006620665458653998673150, 8.639329315174475247163859786412, 10.14363216890449431827631241356

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