Properties

Label 2-252-1.1-c11-0-15
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  − 1.05e4·5-s − 1.68e4·7-s + 4.87e5·11-s − 2.94e4·13-s − 2.33e6·17-s − 1.94e6·19-s − 1.87e7·23-s + 6.27e7·25-s − 1.99e7·29-s − 2.43e8·31-s + 1.77e8·35-s + 5.32e8·37-s + 8.90e8·41-s + 5.15e8·43-s + 2.38e9·47-s + 2.82e8·49-s + 3.59e9·53-s − 5.15e9·55-s + 4.83e9·59-s + 2.98e9·61-s + 3.10e8·65-s + 1.87e10·67-s + 5.13e9·71-s − 3.28e10·73-s − 8.19e9·77-s − 2.03e10·79-s + 1.70e10·83-s + ⋯
L(s)  = 1  − 1.51·5-s − 0.377·7-s + 0.913·11-s − 0.0219·13-s − 0.398·17-s − 0.179·19-s − 0.608·23-s + 1.28·25-s − 0.180·29-s − 1.52·31-s + 0.571·35-s + 1.26·37-s + 1.20·41-s + 0.534·43-s + 1.51·47-s + 0.142·49-s + 1.18·53-s − 1.38·55-s + 0.879·59-s + 0.452·61-s + 0.0332·65-s + 1.69·67-s + 0.337·71-s − 1.85·73-s − 0.345·77-s − 0.742·79-s + 0.474·83-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 \( 1 + 1.05e4T + 4.88e7T^{2} \)
11 \( 1 - 4.87e5T + 2.85e11T^{2} \)
13 \( 1 + 2.94e4T + 1.79e12T^{2} \)
17 \( 1 + 2.33e6T + 3.42e13T^{2} \)
19 \( 1 + 1.94e6T + 1.16e14T^{2} \)
23 \( 1 + 1.87e7T + 9.52e14T^{2} \)
29 \( 1 + 1.99e7T + 1.22e16T^{2} \)
31 \( 1 + 2.43e8T + 2.54e16T^{2} \)
37 \( 1 - 5.32e8T + 1.77e17T^{2} \)
41 \( 1 - 8.90e8T + 5.50e17T^{2} \)
43 \( 1 - 5.15e8T + 9.29e17T^{2} \)
47 \( 1 - 2.38e9T + 2.47e18T^{2} \)
53 \( 1 - 3.59e9T + 9.26e18T^{2} \)
59 \( 1 - 4.83e9T + 3.01e19T^{2} \)
61 \( 1 - 2.98e9T + 4.35e19T^{2} \)
67 \( 1 - 1.87e10T + 1.22e20T^{2} \)
71 \( 1 - 5.13e9T + 2.31e20T^{2} \)
73 \( 1 + 3.28e10T + 3.13e20T^{2} \)
79 \( 1 + 2.03e10T + 7.47e20T^{2} \)
83 \( 1 - 1.70e10T + 1.28e21T^{2} \)
89 \( 1 + 4.06e10T + 2.77e21T^{2} \)
97 \( 1 - 9.26e10T + 7.15e21T^{2} \)
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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

−9.489799215609054040441788266227, −8.642848109634703128078575213345, −7.63387946105224183202334502966, −6.89121284941605057620367017602, −5.71012130317550334942519739132, −4.16072320033572692206982767457, −3.84399257037907854215826262180, −2.47296072030607086807977270049, −0.955745956902430902896537478697, 0, 0.955745956902430902896537478697, 2.47296072030607086807977270049, 3.84399257037907854215826262180, 4.16072320033572692206982767457, 5.71012130317550334942519739132, 6.89121284941605057620367017602, 7.63387946105224183202334502966, 8.642848109634703128078575213345, 9.489799215609054040441788266227

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