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Properties

Label 2-14e2-1.1-c5-0-8
Degree 22
Conductor 196196
Sign −1-1
Analytic cond. 31.435231.4352
Root an. cond. 5.606715.60671
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26·3-s − 16·5-s + 433·9-s + 8·11-s − 684·13-s + 416·15-s + 2.21e3·17-s + 2.69e3·19-s + 3.34e3·23-s − 2.86e3·25-s − 4.94e3·27-s − 3.25e3·29-s − 4.78e3·31-s − 208·33-s − 1.14e4·37-s + 1.77e4·39-s − 1.33e4·41-s − 928·43-s − 6.92e3·45-s − 1.21e3·47-s − 5.76e4·51-s + 1.31e4·53-s − 128·55-s − 7.01e4·57-s − 3.47e4·59-s + 1.03e3·61-s + 1.09e4·65-s + ⋯
L(s)  = 1  − 1.66·3-s − 0.286·5-s + 1.78·9-s + 0.0199·11-s − 1.12·13-s + 0.477·15-s + 1.86·17-s + 1.71·19-s + 1.31·23-s − 0.918·25-s − 1.30·27-s − 0.718·29-s − 0.894·31-s − 0.0332·33-s − 1.37·37-s + 1.87·39-s − 1.24·41-s − 0.0765·43-s − 0.510·45-s − 0.0800·47-s − 3.10·51-s + 0.641·53-s − 0.00570·55-s − 2.85·57-s − 1.29·59-s + 0.0355·61-s + 0.321·65-s + ⋯

Functional equation

Λ(s)=(196s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(196s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 196196    =    22⋅722^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 31.435231.4352
Root analytic conductor: 5.606715.60671
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 196, ( :5/2), −1)(2,\ 196,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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
7 1 1
good3 1+26T+p5T2 1 + 26 T + p^{5} T^{2}
5 1+16T+p5T2 1 + 16 T + p^{5} T^{2}
11 1−8T+p5T2 1 - 8 T + p^{5} T^{2}
13 1+684T+p5T2 1 + 684 T + p^{5} T^{2}
17 1−2218T+p5T2 1 - 2218 T + p^{5} T^{2}
19 1−142pT+p5T2 1 - 142 p T + p^{5} T^{2}
23 1−3344T+p5T2 1 - 3344 T + p^{5} T^{2}
29 1+3254T+p5T2 1 + 3254 T + p^{5} T^{2}
31 1+4788T+p5T2 1 + 4788 T + p^{5} T^{2}
37 1+310pT+p5T2 1 + 310 p T + p^{5} T^{2}
41 1+13350T+p5T2 1 + 13350 T + p^{5} T^{2}
43 1+928T+p5T2 1 + 928 T + p^{5} T^{2}
47 1+1212T+p5T2 1 + 1212 T + p^{5} T^{2}
53 1−13110T+p5T2 1 - 13110 T + p^{5} T^{2}
59 1+34702T+p5T2 1 + 34702 T + p^{5} T^{2}
61 1−1032T+p5T2 1 - 1032 T + p^{5} T^{2}
67 1−10108T+p5T2 1 - 10108 T + p^{5} T^{2}
71 1−62720T+p5T2 1 - 62720 T + p^{5} T^{2}
73 1−18926T+p5T2 1 - 18926 T + p^{5} T^{2}
79 1−11400T+p5T2 1 - 11400 T + p^{5} T^{2}
83 1+88958T+p5T2 1 + 88958 T + p^{5} T^{2}
89 1+19722T+p5T2 1 + 19722 T + p^{5} T^{2}
97 1+17062T+p5T2 1 + 17062 T + p^{5} T^{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

−11.33471956710895150907029557543, −10.25188740449509304852350380857, −9.509433783961479908174097382477, −7.65460118025772319897637611203, −6.98565635072543990461380114714, −5.48558806131182120012246832133, −5.14065714893607491960918807244, −3.45912737019757210542685232567, −1.26148521739396053398035059599, 0, 1.26148521739396053398035059599, 3.45912737019757210542685232567, 5.14065714893607491960918807244, 5.48558806131182120012246832133, 6.98565635072543990461380114714, 7.65460118025772319897637611203, 9.509433783961479908174097382477, 10.25188740449509304852350380857, 11.33471956710895150907029557543

Graph of the ZZ-function along the critical line