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Properties

Label 2-14e2-1.1-c5-0-10
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  − 16·3-s + 16·5-s + 13·9-s − 76·11-s + 880·13-s − 256·15-s − 1.05e3·17-s + 1.93e3·19-s + 936·23-s − 2.86e3·25-s + 3.68e3·27-s − 3.98e3·29-s + 1.56e3·31-s + 1.21e3·33-s + 4.93e3·37-s − 1.40e4·39-s − 1.58e4·41-s − 1.64e4·43-s + 208·45-s − 2.07e4·47-s + 1.68e4·51-s − 3.74e4·53-s − 1.21e3·55-s − 3.09e4·57-s + 2.11e4·59-s − 2.99e3·61-s + 1.40e4·65-s + ⋯
L(s)  = 1  − 1.02·3-s + 0.286·5-s + 0.0534·9-s − 0.189·11-s + 1.44·13-s − 0.293·15-s − 0.886·17-s + 1.23·19-s + 0.368·23-s − 0.918·25-s + 0.971·27-s − 0.879·29-s + 0.293·31-s + 0.194·33-s + 0.592·37-s − 1.48·39-s − 1.47·41-s − 1.35·43-s + 0.0153·45-s − 1.37·47-s + 0.909·51-s − 1.82·53-s − 0.0542·55-s − 1.26·57-s + 0.790·59-s − 0.102·61-s + 0.413·65-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+16T+p5T2 1 + 16 T + p^{5} T^{2}
5 1−16T+p5T2 1 - 16 T + p^{5} T^{2}
11 1+76T+p5T2 1 + 76 T + p^{5} T^{2}
13 1−880T+p5T2 1 - 880 T + p^{5} T^{2}
17 1+1056T+p5T2 1 + 1056 T + p^{5} T^{2}
19 1−1936T+p5T2 1 - 1936 T + p^{5} T^{2}
23 1−936T+p5T2 1 - 936 T + p^{5} T^{2}
29 1+3982T+p5T2 1 + 3982 T + p^{5} T^{2}
31 1−1568T+p5T2 1 - 1568 T + p^{5} T^{2}
37 1−4938T+p5T2 1 - 4938 T + p^{5} T^{2}
41 1+15840T+p5T2 1 + 15840 T + p^{5} T^{2}
43 1+16412T+p5T2 1 + 16412 T + p^{5} T^{2}
47 1+20768T+p5T2 1 + 20768 T + p^{5} T^{2}
53 1+37402T+p5T2 1 + 37402 T + p^{5} T^{2}
59 1−21136T+p5T2 1 - 21136 T + p^{5} T^{2}
61 1+2992T+p5T2 1 + 2992 T + p^{5} T^{2}
67 1+45836T+p5T2 1 + 45836 T + p^{5} T^{2}
71 1+49840T+p5T2 1 + 49840 T + p^{5} T^{2}
73 1+56320T+p5T2 1 + 56320 T + p^{5} T^{2}
79 1−40744T+p5T2 1 - 40744 T + p^{5} T^{2}
83 1−112464T+p5T2 1 - 112464 T + p^{5} T^{2}
89 1−64256T+p5T2 1 - 64256 T + p^{5} T^{2}
97 1+2272T+p5T2 1 + 2272 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

−11.30993833705951049102484630701, −10.33827622522800303758518587013, −9.202464916592899095833117710846, −8.079338207150592065476131412267, −6.65820021598812265171019338656, −5.86798867726540691966244533113, −4.88215768015196385897094500276, −3.33606421234861251947254713755, −1.48732122041019195017557348087, 0, 1.48732122041019195017557348087, 3.33606421234861251947254713755, 4.88215768015196385897094500276, 5.86798867726540691966244533113, 6.65820021598812265171019338656, 8.079338207150592065476131412267, 9.202464916592899095833117710846, 10.33827622522800303758518587013, 11.30993833705951049102484630701

Graph of the ZZ-function along the critical line