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Properties

Label 2-3240-1.1-c1-0-13
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 − 3.27·7-s + 6.27·11-s − 1.27·13-s + 2·17-s + 19-s − 7.27·23-s + 25-s + 6.27·29-s + 6.27·31-s − 3.27·35-s − 10.5·37-s + 7.54·41-s + 4·43-s − 1.27·47-s + 3.72·49-s + 0.725·53-s + 6.27·55-s − 13·59-s + 8.54·61-s − 1.27·65-s + 0.549·67-s − 8.27·71-s + 15.0·73-s − 20.5·77-s + 10.5·79-s − 2.54·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.23·7-s + 1.89·11-s − 0.353·13-s + 0.485·17-s + 0.229·19-s − 1.51·23-s + 0.200·25-s + 1.16·29-s + 1.12·31-s − 0.553·35-s − 1.73·37-s + 1.17·41-s + 0.609·43-s − 0.185·47-s + 0.532·49-s + 0.0995·53-s + 0.846·55-s − 1.69·59-s + 1.09·61-s − 0.158·65-s + 0.0671·67-s − 0.982·71-s + 1.76·73-s − 2.34·77-s + 1.18·79-s − 0.279·83-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 1.9089396841.908939684
L(12)L(\frac12) ≈\approx 1.9089396841.908939684
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+3.27T+7T2 1 + 3.27T + 7T^{2}
11 1−6.27T+11T2 1 - 6.27T + 11T^{2}
13 1+1.27T+13T2 1 + 1.27T + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+7.27T+23T2 1 + 7.27T + 23T^{2}
29 1−6.27T+29T2 1 - 6.27T + 29T^{2}
31 1−6.27T+31T2 1 - 6.27T + 31T^{2}
37 1+10.5T+37T2 1 + 10.5T + 37T^{2}
41 1−7.54T+41T2 1 - 7.54T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+1.27T+47T2 1 + 1.27T + 47T^{2}
53 1−0.725T+53T2 1 - 0.725T + 53T^{2}
59 1+13T+59T2 1 + 13T + 59T^{2}
61 1−8.54T+61T2 1 - 8.54T + 61T^{2}
67 1−0.549T+67T2 1 - 0.549T + 67T^{2}
71 1+8.27T+71T2 1 + 8.27T + 71T^{2}
73 1−15.0T+73T2 1 - 15.0T + 73T^{2}
79 1−10.5T+79T2 1 - 10.5T + 79T^{2}
83 1+2.54T+83T2 1 + 2.54T + 83T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1−16T+97T2 1 - 16T + 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

−8.857449078695155521324121511884, −7.917225906548906442004130191206, −6.93137968145181140621046171731, −6.35870535421854895138558621178, −5.93421836585419308708142466618, −4.73688837916018369756851686823, −3.84430790532800964179544769989, −3.16432154335307708493304060967, −2.03295347094569248608226426374, −0.844460583937210432502758867826, 0.844460583937210432502758867826, 2.03295347094569248608226426374, 3.16432154335307708493304060967, 3.84430790532800964179544769989, 4.73688837916018369756851686823, 5.93421836585419308708142466618, 6.35870535421854895138558621178, 6.93137968145181140621046171731, 7.917225906548906442004130191206, 8.857449078695155521324121511884

Graph of the ZZ-function along the critical line