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Properties

Label 2-1160-1.1-c1-0-26
Degree 22
Conductor 11601160
Sign −1-1
Analytic cond. 9.262649.26264
Root an. cond. 3.043453.04345
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  + 1.70·3-s − 5-s − 0.630·7-s − 0.0783·9-s − 3.70·11-s − 3.07·13-s − 1.70·15-s − 6.04·17-s − 1.36·19-s − 1.07·21-s + 1.70·23-s + 25-s − 5.26·27-s + 29-s − 4.78·31-s − 6.34·33-s + 0.630·35-s − 2.63·37-s − 5.26·39-s + 8.34·41-s + 1.12·43-s + 0.0783·45-s + 10.3·47-s − 6.60·49-s − 10.3·51-s + 9.02·53-s + 3.70·55-s + ⋯
L(s)  = 1  + 0.986·3-s − 0.447·5-s − 0.238·7-s − 0.0261·9-s − 1.11·11-s − 0.853·13-s − 0.441·15-s − 1.46·17-s − 0.314·19-s − 0.235·21-s + 0.356·23-s + 0.200·25-s − 1.01·27-s + 0.185·29-s − 0.859·31-s − 1.10·33-s + 0.106·35-s − 0.432·37-s − 0.842·39-s + 1.30·41-s + 0.171·43-s + 0.0116·45-s + 1.51·47-s − 0.943·49-s − 1.44·51-s + 1.23·53-s + 0.500·55-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11601160    =    23⋅5⋅292^{3} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 9.262649.26264
Root analytic conductor: 3.043453.04345
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1160, ( :1/2), −1)(2,\ 1160,\ (\ :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
5 1+T 1 + T
29 1−T 1 - T
good3 1−1.70T+3T2 1 - 1.70T + 3T^{2}
7 1+0.630T+7T2 1 + 0.630T + 7T^{2}
11 1+3.70T+11T2 1 + 3.70T + 11T^{2}
13 1+3.07T+13T2 1 + 3.07T + 13T^{2}
17 1+6.04T+17T2 1 + 6.04T + 17T^{2}
19 1+1.36T+19T2 1 + 1.36T + 19T^{2}
23 1−1.70T+23T2 1 - 1.70T + 23T^{2}
31 1+4.78T+31T2 1 + 4.78T + 31T^{2}
37 1+2.63T+37T2 1 + 2.63T + 37T^{2}
41 1−8.34T+41T2 1 - 8.34T + 41T^{2}
43 1−1.12T+43T2 1 - 1.12T + 43T^{2}
47 1−10.3T+47T2 1 - 10.3T + 47T^{2}
53 1−9.02T+53T2 1 - 9.02T + 53T^{2}
59 1−5.75T+59T2 1 - 5.75T + 59T^{2}
61 1+9.60T+61T2 1 + 9.60T + 61T^{2}
67 1+8.44T+67T2 1 + 8.44T + 67T^{2}
71 1+9.75T+71T2 1 + 9.75T + 71T^{2}
73 1+4.29T+73T2 1 + 4.29T + 73T^{2}
79 1−6.23T+79T2 1 - 6.23T + 79T^{2}
83 1+11.1T+83T2 1 + 11.1T + 83T^{2}
89 1−2.58T+89T2 1 - 2.58T + 89T^{2}
97 1−13.6T+97T2 1 - 13.6T + 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

−9.051152687093268761748124520537, −8.752577536541371190131994321339, −7.66969100124969520993081286884, −7.25699716096968861738755498117, −6.01581734009807422060920750782, −4.93240679859869155194793535534, −4.00752715486541866783501543151, −2.86580362066059098182257105810, −2.24726801042318342357932078609, 0, 2.24726801042318342357932078609, 2.86580362066059098182257105810, 4.00752715486541866783501543151, 4.93240679859869155194793535534, 6.01581734009807422060920750782, 7.25699716096968861738755498117, 7.66969100124969520993081286884, 8.752577536541371190131994321339, 9.051152687093268761748124520537

Graph of the ZZ-function along the critical line