Properties

Label 2-252-1.1-c11-0-11
Degree $2$
Conductor $252$
Sign $1$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.03e3·5-s + 1.68e4·7-s − 1.56e4·11-s + 1.94e6·13-s − 1.06e7·17-s + 1.61e7·19-s + 1.30e7·23-s + 6.33e5·25-s + 1.29e8·29-s − 2.86e7·31-s + 1.18e8·35-s + 1.36e8·37-s + 5.64e7·41-s − 1.77e8·43-s − 3.18e8·47-s + 2.82e8·49-s + 4.84e9·53-s − 1.10e8·55-s + 3.10e9·59-s + 4.79e9·61-s + 1.36e10·65-s − 5.30e8·67-s − 8.63e9·71-s − 6.97e9·73-s − 2.63e8·77-s − 1.27e9·79-s − 5.73e10·83-s + ⋯
L(s)  = 1  + 1.00·5-s + 0.377·7-s − 0.0293·11-s + 1.44·13-s − 1.81·17-s + 1.49·19-s + 0.423·23-s + 0.0129·25-s + 1.17·29-s − 0.179·31-s + 0.380·35-s + 0.323·37-s + 0.0761·41-s − 0.184·43-s − 0.202·47-s + 0.142·49-s + 1.59·53-s − 0.0295·55-s + 0.565·59-s + 0.727·61-s + 1.45·65-s − 0.0479·67-s − 0.568·71-s − 0.393·73-s − 0.0110·77-s − 0.0465·79-s − 1.59·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(3.516245598\)
\(L(\frac12)\) \(\approx\) \(3.516245598\)
\(L(\frac{13}{2})\) 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 \( 1 - 1.68e4T \)
good5 \( 1 - 7.03e3T + 4.88e7T^{2} \)
11 \( 1 + 1.56e4T + 2.85e11T^{2} \)
13 \( 1 - 1.94e6T + 1.79e12T^{2} \)
17 \( 1 + 1.06e7T + 3.42e13T^{2} \)
19 \( 1 - 1.61e7T + 1.16e14T^{2} \)
23 \( 1 - 1.30e7T + 9.52e14T^{2} \)
29 \( 1 - 1.29e8T + 1.22e16T^{2} \)
31 \( 1 + 2.86e7T + 2.54e16T^{2} \)
37 \( 1 - 1.36e8T + 1.77e17T^{2} \)
41 \( 1 - 5.64e7T + 5.50e17T^{2} \)
43 \( 1 + 1.77e8T + 9.29e17T^{2} \)
47 \( 1 + 3.18e8T + 2.47e18T^{2} \)
53 \( 1 - 4.84e9T + 9.26e18T^{2} \)
59 \( 1 - 3.10e9T + 3.01e19T^{2} \)
61 \( 1 - 4.79e9T + 4.35e19T^{2} \)
67 \( 1 + 5.30e8T + 1.22e20T^{2} \)
71 \( 1 + 8.63e9T + 2.31e20T^{2} \)
73 \( 1 + 6.97e9T + 3.13e20T^{2} \)
79 \( 1 + 1.27e9T + 7.47e20T^{2} \)
83 \( 1 + 5.73e10T + 1.28e21T^{2} \)
89 \( 1 - 1.79e10T + 2.77e21T^{2} \)
97 \( 1 - 6.10e10T + 7.15e21T^{2} \)
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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

−10.10263607575708284851610262517, −9.098553091323407213883680216735, −8.404071895533154949211307736165, −7.04311363326440657142106828832, −6.14244569591158315702360565331, −5.25458407180110899276045880905, −4.10618994457305238771281494580, −2.79860815894972162949346007972, −1.74019810520129965121816788873, −0.830556214051530284741164331108, 0.830556214051530284741164331108, 1.74019810520129965121816788873, 2.79860815894972162949346007972, 4.10618994457305238771281494580, 5.25458407180110899276045880905, 6.14244569591158315702360565331, 7.04311363326440657142106828832, 8.404071895533154949211307736165, 9.098553091323407213883680216735, 10.10263607575708284851610262517

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