Properties

Label 2-252-1.1-c11-0-12
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.05e3·5-s − 1.68e4·7-s − 1.01e6·11-s − 2.12e6·13-s + 5.48e6·17-s + 2.07e7·19-s + 5.46e7·23-s + 3.31e7·25-s + 1.42e8·29-s + 1.72e8·31-s + 1.52e8·35-s − 6.75e8·37-s − 9.09e7·41-s − 4.27e8·43-s − 2.38e8·47-s + 2.82e8·49-s + 1.72e9·53-s + 9.16e9·55-s + 5.53e9·59-s − 3.08e9·61-s + 1.92e10·65-s − 9.55e9·67-s + 1.00e10·71-s − 2.52e10·73-s + 1.70e10·77-s + 2.70e10·79-s − 6.76e10·83-s + ⋯
L(s)  = 1  − 1.29·5-s − 0.377·7-s − 1.89·11-s − 1.58·13-s + 0.937·17-s + 1.92·19-s + 1.77·23-s + 0.679·25-s + 1.29·29-s + 1.08·31-s + 0.489·35-s − 1.60·37-s − 0.122·41-s − 0.443·43-s − 0.151·47-s + 0.142·49-s + 0.567·53-s + 2.45·55-s + 1.00·59-s − 0.468·61-s + 2.05·65-s − 0.864·67-s + 0.662·71-s − 1.42·73-s + 0.716·77-s + 0.988·79-s − 1.88·83-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: \(1\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 9.05e3T + 4.88e7T^{2} \)
11 \( 1 + 1.01e6T + 2.85e11T^{2} \)
13 \( 1 + 2.12e6T + 1.79e12T^{2} \)
17 \( 1 - 5.48e6T + 3.42e13T^{2} \)
19 \( 1 - 2.07e7T + 1.16e14T^{2} \)
23 \( 1 - 5.46e7T + 9.52e14T^{2} \)
29 \( 1 - 1.42e8T + 1.22e16T^{2} \)
31 \( 1 - 1.72e8T + 2.54e16T^{2} \)
37 \( 1 + 6.75e8T + 1.77e17T^{2} \)
41 \( 1 + 9.09e7T + 5.50e17T^{2} \)
43 \( 1 + 4.27e8T + 9.29e17T^{2} \)
47 \( 1 + 2.38e8T + 2.47e18T^{2} \)
53 \( 1 - 1.72e9T + 9.26e18T^{2} \)
59 \( 1 - 5.53e9T + 3.01e19T^{2} \)
61 \( 1 + 3.08e9T + 4.35e19T^{2} \)
67 \( 1 + 9.55e9T + 1.22e20T^{2} \)
71 \( 1 - 1.00e10T + 2.31e20T^{2} \)
73 \( 1 + 2.52e10T + 3.13e20T^{2} \)
79 \( 1 - 2.70e10T + 7.47e20T^{2} \)
83 \( 1 + 6.76e10T + 1.28e21T^{2} \)
89 \( 1 + 1.58e10T + 2.77e21T^{2} \)
97 \( 1 - 5.58e10T + 7.15e21T^{2} \)
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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

−9.838020656842419148710388477153, −8.456764880601793510111973796454, −7.53560708923942367278113023731, −7.15790410959097869438230701853, −5.29740788684191242981675906055, −4.82408777165753981930246065791, −3.21328685010183656383061181616, −2.78408512742469810732919338254, −0.877239567260961485812388975802, 0, 0.877239567260961485812388975802, 2.78408512742469810732919338254, 3.21328685010183656383061181616, 4.82408777165753981930246065791, 5.29740788684191242981675906055, 7.15790410959097869438230701853, 7.53560708923942367278113023731, 8.456764880601793510111973796454, 9.838020656842419148710388477153

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