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Properties

Label 2-252-7.4-c11-0-27
Degree $2$
Conductor $252$
Sign $0.234 + 0.972i$
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  + (−84.6 − 146. i)5-s + (−9.85e3 + 4.33e4i)7-s + (1.43e5 − 2.49e5i)11-s + 2.48e6·13-s + (3.58e6 − 6.20e6i)17-s + (−1.63e6 − 2.83e6i)19-s + (−1.56e7 − 2.71e7i)23-s + (2.43e7 − 4.22e7i)25-s − 5.63e7·29-s + (−8.04e7 + 1.39e8i)31-s + (7.18e6 − 2.22e6i)35-s + (1.48e8 + 2.57e8i)37-s − 1.39e9·41-s − 4.76e8·43-s + (1.19e9 + 2.06e9i)47-s + ⋯
L(s)  = 1  + (−0.0121 − 0.0209i)5-s + (−0.221 + 0.975i)7-s + (0.269 − 0.466i)11-s + 1.85·13-s + (0.611 − 1.05i)17-s + (−0.151 − 0.262i)19-s + (−0.507 − 0.878i)23-s + (0.499 − 0.865i)25-s − 0.509·29-s + (−0.504 + 0.874i)31-s + (0.0231 − 0.00716i)35-s + (0.352 + 0.611i)37-s − 1.88·41-s − 0.494·43-s + (0.758 + 1.31i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.234 + 0.972i$
Analytic conductor: \(193.622\)
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: \((2,\ 252,\ (\ :11/2),\ 0.234 + 0.972i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.934258009\)
\(L(\frac12)\) \(\approx\) \(1.934258009\)
\(L(\frac{13}{2})\) 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 \( 1 + (9.85e3 - 4.33e4i)T \)
good5 \( 1 + (84.6 + 146. i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-1.43e5 + 2.49e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 - 2.48e6T + 1.79e12T^{2} \)
17 \( 1 + (-3.58e6 + 6.20e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (1.63e6 + 2.83e6i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (1.56e7 + 2.71e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + 5.63e7T + 1.22e16T^{2} \)
31 \( 1 + (8.04e7 - 1.39e8i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (-1.48e8 - 2.57e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 1.39e9T + 5.50e17T^{2} \)
43 \( 1 + 4.76e8T + 9.29e17T^{2} \)
47 \( 1 + (-1.19e9 - 2.06e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-2.34e9 + 4.05e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (-8.21e8 + 1.42e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (1.55e9 + 2.69e9i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (5.33e9 - 9.24e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 2.59e10T + 2.31e20T^{2} \)
73 \( 1 + (2.45e8 - 4.24e8i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (4.61e9 + 7.98e9i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 3.48e10T + 1.28e21T^{2} \)
89 \( 1 + (1.81e10 + 3.15e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 1.51e11T + 7.15e21T^{2} \)
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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

−9.845014615109686574437527702988, −8.742521783020600593491232652280, −8.334181338630285938783230797170, −6.76478255328104237502629842360, −6.02114106012836271135293769786, −5.03117685958867322140321970888, −3.66866096959285959237141379180, −2.78628047916272873471419167270, −1.49970955427997187958011239820, −0.38892313228446263877155911542, 1.01706927005932234262741629198, 1.76010811611861681082616926964, 3.64239914988448272840406892449, 3.85808204916374543469484898930, 5.48713899740476899764384660330, 6.40685012679210473492537300940, 7.41534294706213828122002240600, 8.343064707046950935949711196535, 9.399979884679302802629573279946, 10.42260247063581530250423368781

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