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Properties

Label 2-40e2-1.1-c1-0-9
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
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  − 2.23·3-s + 4.47·7-s + 2.00·9-s + 2.23·11-s + 4·13-s + 7·17-s − 6.70·19-s − 10.0·21-s − 4.47·23-s + 2.23·27-s − 4.47·31-s − 5.00·33-s + 2·37-s − 8.94·39-s + 5·41-s − 8.94·47-s + 13.0·49-s − 15.6·51-s + 6·53-s + 15.0·57-s + 8.94·59-s − 10·61-s + 8.94·63-s + 2.23·67-s + 10.0·69-s + 8.94·71-s + 9·73-s + ⋯
L(s)  = 1  − 1.29·3-s + 1.69·7-s + 0.666·9-s + 0.674·11-s + 1.10·13-s + 1.69·17-s − 1.53·19-s − 2.18·21-s − 0.932·23-s + 0.430·27-s − 0.803·31-s − 0.870·33-s + 0.328·37-s − 1.43·39-s + 0.780·41-s − 1.30·47-s + 1.85·49-s − 2.19·51-s + 0.824·53-s + 1.98·57-s + 1.16·59-s − 1.28·61-s + 1.12·63-s + 0.273·67-s + 1.20·69-s + 1.06·71-s + 1.05·73-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 1)(2,\ 1600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4382145821.438214582
L(12)L(\frac12) ≈\approx 1.4382145821.438214582
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 1
good3 1+2.23T+3T2 1 + 2.23T + 3T^{2}
7 1−4.47T+7T2 1 - 4.47T + 7T^{2}
11 1−2.23T+11T2 1 - 2.23T + 11T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
17 1−7T+17T2 1 - 7T + 17T^{2}
19 1+6.70T+19T2 1 + 6.70T + 19T^{2}
23 1+4.47T+23T2 1 + 4.47T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1+4.47T+31T2 1 + 4.47T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−5T+41T2 1 - 5T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+8.94T+47T2 1 + 8.94T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−8.94T+59T2 1 - 8.94T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1−2.23T+67T2 1 - 2.23T + 67T^{2}
71 1−8.94T+71T2 1 - 8.94T + 71T^{2}
73 1−9T+73T2 1 - 9T + 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1−11.1T+83T2 1 - 11.1T + 83T^{2}
89 1+5T+89T2 1 + 5T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−9.474626111714987005105452428289, −8.351852504600398419862760046948, −7.996825105458390711585015727874, −6.81597589739661095394337101429, −5.97216720530962660687726711371, −5.44232202253705255954242354875, −4.52793571040102260089164909665, −3.73742767373973948123899984573, −1.90743131121157376759686576618, −0.971631826011971749588523265304, 0.971631826011971749588523265304, 1.90743131121157376759686576618, 3.73742767373973948123899984573, 4.52793571040102260089164909665, 5.44232202253705255954242354875, 5.97216720530962660687726711371, 6.81597589739661095394337101429, 7.996825105458390711585015727874, 8.351852504600398419862760046948, 9.474626111714987005105452428289

Graph of the ZZ-function along the critical line