Properties

Label 2-252-1.1-c11-0-2
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  − 1.00e4·5-s − 1.68e4·7-s + 1.73e5·11-s + 3.17e5·13-s − 7.04e6·17-s + 4.16e6·19-s − 1.61e7·23-s + 5.17e7·25-s − 4.07e7·29-s − 7.53e7·31-s + 1.68e8·35-s − 2.10e8·37-s − 9.22e8·41-s − 1.14e9·43-s − 1.65e9·47-s + 2.82e8·49-s − 2.92e9·53-s − 1.74e9·55-s + 4.21e9·59-s + 1.60e8·61-s − 3.18e9·65-s − 1.31e10·67-s − 9.84e9·71-s + 1.36e10·73-s − 2.92e9·77-s + 2.37e10·79-s − 3.65e10·83-s + ⋯
L(s)  = 1  − 1.43·5-s − 0.377·7-s + 0.325·11-s + 0.236·13-s − 1.20·17-s + 0.385·19-s − 0.522·23-s + 1.05·25-s − 0.368·29-s − 0.472·31-s + 0.542·35-s − 0.498·37-s − 1.24·41-s − 1.19·43-s − 1.05·47-s + 0.142·49-s − 0.959·53-s − 0.467·55-s + 0.768·59-s + 0.0243·61-s − 0.340·65-s − 1.18·67-s − 0.647·71-s + 0.771·73-s − 0.123·77-s + 0.869·79-s − 1.01·83-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

\(L(6)\) \(\approx\) \(0.5344088640\)
\(L(\frac12)\) \(\approx\) \(0.5344088640\)
\(L(\frac{13}{2})\) 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 \( 1 + 1.00e4T + 4.88e7T^{2} \)
11 \( 1 - 1.73e5T + 2.85e11T^{2} \)
13 \( 1 - 3.17e5T + 1.79e12T^{2} \)
17 \( 1 + 7.04e6T + 3.42e13T^{2} \)
19 \( 1 - 4.16e6T + 1.16e14T^{2} \)
23 \( 1 + 1.61e7T + 9.52e14T^{2} \)
29 \( 1 + 4.07e7T + 1.22e16T^{2} \)
31 \( 1 + 7.53e7T + 2.54e16T^{2} \)
37 \( 1 + 2.10e8T + 1.77e17T^{2} \)
41 \( 1 + 9.22e8T + 5.50e17T^{2} \)
43 \( 1 + 1.14e9T + 9.29e17T^{2} \)
47 \( 1 + 1.65e9T + 2.47e18T^{2} \)
53 \( 1 + 2.92e9T + 9.26e18T^{2} \)
59 \( 1 - 4.21e9T + 3.01e19T^{2} \)
61 \( 1 - 1.60e8T + 4.35e19T^{2} \)
67 \( 1 + 1.31e10T + 1.22e20T^{2} \)
71 \( 1 + 9.84e9T + 2.31e20T^{2} \)
73 \( 1 - 1.36e10T + 3.13e20T^{2} \)
79 \( 1 - 2.37e10T + 7.47e20T^{2} \)
83 \( 1 + 3.65e10T + 1.28e21T^{2} \)
89 \( 1 + 6.40e10T + 2.77e21T^{2} \)
97 \( 1 - 1.39e11T + 7.15e21T^{2} \)
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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

−10.15964544219455807077473368857, −8.988502045909092537893935693726, −8.194969720200262596029145303800, −7.22177813009100600709756149294, −6.38292667882757868701775618271, −4.94119778473463271490679833233, −3.94292509769771422398823362129, −3.21147592076841682154357264741, −1.72996657924751573306965872966, −0.30218417808025772955629123448, 0.30218417808025772955629123448, 1.72996657924751573306965872966, 3.21147592076841682154357264741, 3.94292509769771422398823362129, 4.94119778473463271490679833233, 6.38292667882757868701775618271, 7.22177813009100600709756149294, 8.194969720200262596029145303800, 8.988502045909092537893935693726, 10.15964544219455807077473368857

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