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Properties

Label 2-3240-1.1-c1-0-7
Degree 22
Conductor 32403240
Sign 11
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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  − 5-s − 2.37·7-s + 3.37·11-s + 2.37·13-s − 4.74·17-s + 19-s + 0.372·23-s + 25-s + 3.37·29-s − 6.11·31-s + 2.37·35-s + 6·37-s − 11.7·41-s + 6.74·43-s + 3.62·47-s − 1.37·49-s + 7.11·53-s − 3.37·55-s − 5·59-s + 1.25·61-s − 2.37·65-s + 10.7·67-s − 1.37·71-s + 3.25·73-s − 8·77-s + 8.74·79-s + 10·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.896·7-s + 1.01·11-s + 0.657·13-s − 1.15·17-s + 0.229·19-s + 0.0776·23-s + 0.200·25-s + 0.626·29-s − 1.09·31-s + 0.400·35-s + 0.986·37-s − 1.83·41-s + 1.02·43-s + 0.529·47-s − 0.196·49-s + 0.977·53-s − 0.454·55-s − 0.650·59-s + 0.160·61-s − 0.294·65-s + 1.31·67-s − 0.162·71-s + 0.381·73-s − 0.911·77-s + 0.983·79-s + 1.09·83-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 11
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 1)(2,\ 3240,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4541976691.454197669
L(12)L(\frac12) ≈\approx 1.4541976691.454197669
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 1
5 1+T 1 + T
good7 1+2.37T+7T2 1 + 2.37T + 7T^{2}
11 1−3.37T+11T2 1 - 3.37T + 11T^{2}
13 1−2.37T+13T2 1 - 2.37T + 13T^{2}
17 1+4.74T+17T2 1 + 4.74T + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1−0.372T+23T2 1 - 0.372T + 23T^{2}
29 1−3.37T+29T2 1 - 3.37T + 29T^{2}
31 1+6.11T+31T2 1 + 6.11T + 31T^{2}
37 1−6T+37T2 1 - 6T + 37T^{2}
41 1+11.7T+41T2 1 + 11.7T + 41T^{2}
43 1−6.74T+43T2 1 - 6.74T + 43T^{2}
47 1−3.62T+47T2 1 - 3.62T + 47T^{2}
53 1−7.11T+53T2 1 - 7.11T + 53T^{2}
59 1+5T+59T2 1 + 5T + 59T^{2}
61 1−1.25T+61T2 1 - 1.25T + 61T^{2}
67 1−10.7T+67T2 1 - 10.7T + 67T^{2}
71 1+1.37T+71T2 1 + 1.37T + 71T^{2}
73 1−3.25T+73T2 1 - 3.25T + 73T^{2}
79 1−8.74T+79T2 1 - 8.74T + 79T^{2}
83 1−10T+83T2 1 - 10T + 83T^{2}
89 1+1.37T+89T2 1 + 1.37T + 89T^{2}
97 1−6.74T+97T2 1 - 6.74T + 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

−8.823515612491441912882040047812, −7.944951697660732035604283625654, −6.95279808740597754452538918287, −6.55361472679277518570412389505, −5.77261846961797279458684752678, −4.66227671868133182889211176051, −3.86123229693247741638981815478, −3.24813357152975580898265374995, −2.05151696499587778666880325446, −0.71987161036480443378039294940, 0.71987161036480443378039294940, 2.05151696499587778666880325446, 3.24813357152975580898265374995, 3.86123229693247741638981815478, 4.66227671868133182889211176051, 5.77261846961797279458684752678, 6.55361472679277518570412389505, 6.95279808740597754452538918287, 7.944951697660732035604283625654, 8.823515612491441912882040047812

Graph of the ZZ-function along the critical line