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Properties

Label 2-6936-1.1-c1-0-101
Degree 22
Conductor 69366936
Sign −1-1
Analytic cond. 55.384255.3842
Root an. cond. 7.442057.44205
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.850·5-s + 1.73·7-s + 9-s + 0.361·11-s + 4.60·13-s + 0.850·15-s − 1.44·19-s − 1.73·21-s + 1.05·23-s − 4.27·25-s − 27-s − 2.88·29-s − 4.17·31-s − 0.361·33-s − 1.47·35-s − 6.58·37-s − 4.60·39-s − 4.23·41-s − 5.22·43-s − 0.850·45-s − 2.60·47-s − 3.99·49-s + 1.61·53-s − 0.307·55-s + 1.44·57-s + 10.9·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.380·5-s + 0.655·7-s + 0.333·9-s + 0.109·11-s + 1.27·13-s + 0.219·15-s − 0.331·19-s − 0.378·21-s + 0.219·23-s − 0.855·25-s − 0.192·27-s − 0.534·29-s − 0.749·31-s − 0.0629·33-s − 0.249·35-s − 1.08·37-s − 0.737·39-s − 0.661·41-s − 0.796·43-s − 0.126·45-s − 0.379·47-s − 0.570·49-s + 0.221·53-s − 0.0415·55-s + 0.191·57-s + 1.42·59-s + ⋯

Functional equation

Λ(s)=(6936s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(6936s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 69366936    =    23⋅3⋅1722^{3} \cdot 3 \cdot 17^{2}
Sign: −1-1
Analytic conductor: 55.384255.3842
Root analytic conductor: 7.442057.44205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 6936, ( :1/2), −1)(2,\ 6936,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
17 1 1
good5 1+0.850T+5T2 1 + 0.850T + 5T^{2}
7 1−1.73T+7T2 1 - 1.73T + 7T^{2}
11 1−0.361T+11T2 1 - 0.361T + 11T^{2}
13 1−4.60T+13T2 1 - 4.60T + 13T^{2}
19 1+1.44T+19T2 1 + 1.44T + 19T^{2}
23 1−1.05T+23T2 1 - 1.05T + 23T^{2}
29 1+2.88T+29T2 1 + 2.88T + 29T^{2}
31 1+4.17T+31T2 1 + 4.17T + 31T^{2}
37 1+6.58T+37T2 1 + 6.58T + 37T^{2}
41 1+4.23T+41T2 1 + 4.23T + 41T^{2}
43 1+5.22T+43T2 1 + 5.22T + 43T^{2}
47 1+2.60T+47T2 1 + 2.60T + 47T^{2}
53 1−1.61T+53T2 1 - 1.61T + 53T^{2}
59 1−10.9T+59T2 1 - 10.9T + 59T^{2}
61 1+9.62T+61T2 1 + 9.62T + 61T^{2}
67 1−8.12T+67T2 1 - 8.12T + 67T^{2}
71 1−1.13T+71T2 1 - 1.13T + 71T^{2}
73 1−3.95T+73T2 1 - 3.95T + 73T^{2}
79 1−1.86T+79T2 1 - 1.86T + 79T^{2}
83 1+11.2T+83T2 1 + 11.2T + 83T^{2}
89 1−14.0T+89T2 1 - 14.0T + 89T^{2}
97 1+15.5T+97T2 1 + 15.5T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−7.60888042278600510277150025695, −6.85936792172869664106517934840, −6.20478086952835101001241386935, −5.45156665354287318402377777428, −4.84689652015228420574588887643, −3.90498262246230633721141415271, −3.45631622667735741308635650223, −2.03483972912618317564268105069, −1.29686502255034160320028069763, 0, 1.29686502255034160320028069763, 2.03483972912618317564268105069, 3.45631622667735741308635650223, 3.90498262246230633721141415271, 4.84689652015228420574588887643, 5.45156665354287318402377777428, 6.20478086952835101001241386935, 6.85936792172869664106517934840, 7.60888042278600510277150025695

Graph of the ZZ-function along the critical line