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Properties

Label 2-4655-1.1-c1-0-64
Degree 22
Conductor 46554655
Sign 11
Analytic cond. 37.170337.1703
Root an. cond. 6.096756.09675
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.09·2-s + 0.170·3-s − 0.796·4-s + 5-s + 0.187·6-s − 3.06·8-s − 2.97·9-s + 1.09·10-s − 5.62·11-s − 0.135·12-s + 6.22·13-s + 0.170·15-s − 1.77·16-s + 0.620·17-s − 3.25·18-s − 19-s − 0.796·20-s − 6.16·22-s − 3.46·23-s − 0.523·24-s + 25-s + 6.83·26-s − 1.01·27-s + 1.28·29-s + 0.187·30-s + 0.00111·31-s + 4.19·32-s + ⋯
L(s)  = 1  + 0.775·2-s + 0.0985·3-s − 0.398·4-s + 0.447·5-s + 0.0764·6-s − 1.08·8-s − 0.990·9-s + 0.346·10-s − 1.69·11-s − 0.0392·12-s + 1.72·13-s + 0.0440·15-s − 0.443·16-s + 0.150·17-s − 0.768·18-s − 0.229·19-s − 0.178·20-s − 1.31·22-s − 0.721·23-s − 0.106·24-s + 0.200·25-s + 1.34·26-s − 0.196·27-s + 0.238·29-s + 0.0341·30-s + 0.000199·31-s + 0.740·32-s + ⋯

Functional equation

Λ(s)=(4655s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4655s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4655 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46554655    =    5⋅72⋅195 \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 37.170337.1703
Root analytic conductor: 6.096756.09675
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4655, ( :1/2), 1)(2,\ 4655,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0302570562.030257056
L(12)L(\frac12) ≈\approx 2.0302570562.030257056
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)
bad5 1−T 1 - T
7 1 1
19 1+T 1 + T
good2 1−1.09T+2T2 1 - 1.09T + 2T^{2}
3 1−0.170T+3T2 1 - 0.170T + 3T^{2}
11 1+5.62T+11T2 1 + 5.62T + 11T^{2}
13 1−6.22T+13T2 1 - 6.22T + 13T^{2}
17 1−0.620T+17T2 1 - 0.620T + 17T^{2}
23 1+3.46T+23T2 1 + 3.46T + 23T^{2}
29 1−1.28T+29T2 1 - 1.28T + 29T^{2}
31 1−0.00111T+31T2 1 - 0.00111T + 31T^{2}
37 1−8.59T+37T2 1 - 8.59T + 37T^{2}
41 1−11.9T+41T2 1 - 11.9T + 41T^{2}
43 1−2.87T+43T2 1 - 2.87T + 43T^{2}
47 1+7.17T+47T2 1 + 7.17T + 47T^{2}
53 1+0.0309T+53T2 1 + 0.0309T + 53T^{2}
59 1−2.44T+59T2 1 - 2.44T + 59T^{2}
61 1+6.52T+61T2 1 + 6.52T + 61T^{2}
67 1−9.19T+67T2 1 - 9.19T + 67T^{2}
71 1−10.0T+71T2 1 - 10.0T + 71T^{2}
73 1−3.20T+73T2 1 - 3.20T + 73T^{2}
79 1+11.0T+79T2 1 + 11.0T + 79T^{2}
83 1−13.3T+83T2 1 - 13.3T + 83T^{2}
89 1−6.43T+89T2 1 - 6.43T + 89T^{2}
97 1−10.8T+97T2 1 - 10.8T + 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

−8.127182027048399542613013826557, −7.952659987242963936593980760358, −6.44253073746498780455147622270, −5.90332907609704335679197355850, −5.48365467182043022271334753619, −4.64081789875964345394495428610, −3.77214850821093148014321227695, −2.98422454209518442158052958056, −2.29485666211271276528479784846, −0.67973490523716963338330891277, 0.67973490523716963338330891277, 2.29485666211271276528479784846, 2.98422454209518442158052958056, 3.77214850821093148014321227695, 4.64081789875964345394495428610, 5.48365467182043022271334753619, 5.90332907609704335679197355850, 6.44253073746498780455147622270, 7.952659987242963936593980760358, 8.127182027048399542613013826557

Graph of the ZZ-function along the critical line