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Properties

Label 2-9405-1.1-c1-0-138
Degree 22
Conductor 94059405
Sign −1-1
Analytic cond. 75.099375.0993
Root an. cond. 8.665988.66598
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  − 2.14·2-s + 2.59·4-s − 5-s − 1.53·7-s − 1.27·8-s + 2.14·10-s − 11-s − 1.14·13-s + 3.29·14-s − 2.45·16-s − 3.14·17-s + 19-s − 2.59·20-s + 2.14·22-s + 5.10·23-s + 25-s + 2.45·26-s − 3.98·28-s − 2.99·29-s + 0.460·31-s + 7.81·32-s + 6.74·34-s + 1.53·35-s − 7.62·37-s − 2.14·38-s + 1.27·40-s + 1.79·41-s + ⋯
L(s)  = 1  − 1.51·2-s + 1.29·4-s − 0.447·5-s − 0.580·7-s − 0.450·8-s + 0.677·10-s − 0.301·11-s − 0.317·13-s + 0.879·14-s − 0.614·16-s − 0.762·17-s + 0.229·19-s − 0.580·20-s + 0.456·22-s + 1.06·23-s + 0.200·25-s + 0.481·26-s − 0.752·28-s − 0.556·29-s + 0.0827·31-s + 1.38·32-s + 1.15·34-s + 0.259·35-s − 1.25·37-s − 0.347·38-s + 0.201·40-s + 0.280·41-s + ⋯

Functional equation

Λ(s)=(9405s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9405s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 94059405    =    32⋅5⋅11⋅193^{2} \cdot 5 \cdot 11 \cdot 19
Sign: −1-1
Analytic conductor: 75.099375.0993
Root analytic conductor: 8.665988.66598
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9405, ( :1/2), −1)(2,\ 9405,\ (\ :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)
bad3 1 1
5 1+T 1 + T
11 1+T 1 + T
19 1−T 1 - T
good2 1+2.14T+2T2 1 + 2.14T + 2T^{2}
7 1+1.53T+7T2 1 + 1.53T + 7T^{2}
13 1+1.14T+13T2 1 + 1.14T + 13T^{2}
17 1+3.14T+17T2 1 + 3.14T + 17T^{2}
23 1−5.10T+23T2 1 - 5.10T + 23T^{2}
29 1+2.99T+29T2 1 + 2.99T + 29T^{2}
31 1−0.460T+31T2 1 - 0.460T + 31T^{2}
37 1+7.62T+37T2 1 + 7.62T + 37T^{2}
41 1−1.79T+41T2 1 - 1.79T + 41T^{2}
43 1−10.1T+43T2 1 - 10.1T + 43T^{2}
47 1−0.0901T+47T2 1 - 0.0901T + 47T^{2}
53 1+2.98T+53T2 1 + 2.98T + 53T^{2}
59 1−7.98T+59T2 1 - 7.98T + 59T^{2}
61 1+11.9T+61T2 1 + 11.9T + 61T^{2}
67 1−9.20T+67T2 1 - 9.20T + 67T^{2}
71 1−10.2T+71T2 1 - 10.2T + 71T^{2}
73 1−12.0T+73T2 1 - 12.0T + 73T^{2}
79 1−4.27T+79T2 1 - 4.27T + 79T^{2}
83 1−3.17T+83T2 1 - 3.17T + 83T^{2}
89 1+15.5T+89T2 1 + 15.5T + 89T^{2}
97 1+17.1T+97T2 1 + 17.1T + 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

−7.43399010756401432432975473514, −6.97768306421645791196465566894, −6.37920855863236867739757126426, −5.34894813617305951356075150396, −4.59778892218138200158884974455, −3.66759039019303880854685009147, −2.77982209120409767736875232490, −1.99753903194109081320447012897, −0.884428978131899938845889791585, 0, 0.884428978131899938845889791585, 1.99753903194109081320447012897, 2.77982209120409767736875232490, 3.66759039019303880854685009147, 4.59778892218138200158884974455, 5.34894813617305951356075150396, 6.37920855863236867739757126426, 6.97768306421645791196465566894, 7.43399010756401432432975473514

Graph of the ZZ-function along the critical line