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Properties

Label 2-14-1.1-c7-0-0
Degree 22
Conductor 1414
Sign 11
Analytic cond. 4.373394.37339
Root an. cond. 2.091262.09126
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 82·3-s + 64·4-s + 448·5-s + 656·6-s − 343·7-s − 512·8-s + 4.53e3·9-s − 3.58e3·10-s + 2.40e3·11-s − 5.24e3·12-s + 7.11e3·13-s + 2.74e3·14-s − 3.67e4·15-s + 4.09e3·16-s + 2.48e3·17-s − 3.62e4·18-s + 3.64e4·19-s + 2.86e4·20-s + 2.81e4·21-s − 1.92e4·22-s − 1.28e4·23-s + 4.19e4·24-s + 1.22e5·25-s − 5.69e4·26-s − 1.92e5·27-s − 2.19e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.75·3-s + 1/2·4-s + 1.60·5-s + 1.23·6-s − 0.377·7-s − 0.353·8-s + 2.07·9-s − 1.13·10-s + 0.545·11-s − 0.876·12-s + 0.898·13-s + 0.267·14-s − 2.81·15-s + 1/4·16-s + 0.122·17-s − 1.46·18-s + 1.22·19-s + 0.801·20-s + 0.662·21-s − 0.385·22-s − 0.220·23-s + 0.619·24-s + 1.56·25-s − 0.635·26-s − 1.88·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(14s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(14s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1414    =    2⋅72 \cdot 7
Sign: 11
Analytic conductor: 4.373394.37339
Root analytic conductor: 2.091262.09126
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 14, ( :7/2), 1)(2,\ 14,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.83079003830.8307900383
L(12)L(\frac12) ≈\approx 0.83079003830.8307900383
L(92)L(\frac{9}{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+p3T 1 + p^{3} T
7 1+p3T 1 + p^{3} T
good3 1+82T+p7T2 1 + 82 T + p^{7} T^{2}
5 1−448T+p7T2 1 - 448 T + p^{7} T^{2}
11 1−2408T+p7T2 1 - 2408 T + p^{7} T^{2}
13 1−7116T+p7T2 1 - 7116 T + p^{7} T^{2}
17 1−2486T+p7T2 1 - 2486 T + p^{7} T^{2}
19 1−36482T+p7T2 1 - 36482 T + p^{7} T^{2}
23 1+560pT+p7T2 1 + 560 p T + p^{7} T^{2}
29 1+88094T+p7T2 1 + 88094 T + p^{7} T^{2}
31 1−282636T+p7T2 1 - 282636 T + p^{7} T^{2}
37 1+214534T+p7T2 1 + 214534 T + p^{7} T^{2}
41 1+140874T+p7T2 1 + 140874 T + p^{7} T^{2}
43 1−848pT+p7T2 1 - 848 p T + p^{7} T^{2}
47 1−716868T+p7T2 1 - 716868 T + p^{7} T^{2}
53 1+56946T+p7T2 1 + 56946 T + p^{7} T^{2}
59 1+2149862T+p7T2 1 + 2149862 T + p^{7} T^{2}
61 1−3084360T+p7T2 1 - 3084360 T + p^{7} T^{2}
67 1+3034364T+p7T2 1 + 3034364 T + p^{7} T^{2}
71 1+106624T+p7T2 1 + 106624 T + p^{7} T^{2}
73 1−988930T+p7T2 1 - 988930 T + p^{7} T^{2}
79 1−3415896T+p7T2 1 - 3415896 T + p^{7} T^{2}
83 1+15142T+p7T2 1 + 15142 T + p^{7} T^{2}
89 1−174810T+p7T2 1 - 174810 T + p^{7} T^{2}
97 1−13506790T+p7T2 1 - 13506790 T + p^{7} 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

−17.73610505753424173474109583908, −17.02950834184976504205130192613, −15.96849769400455349787404246440, −13.56339706571102621655797947069, −11.97413054261582370033651680060, −10.57689597773280735553669507498, −9.531843796425403344433679608492, −6.59539017034699864580637435114, −5.62616050802270837992013920583, −1.19385621467107446928197029499, 1.19385621467107446928197029499, 5.62616050802270837992013920583, 6.59539017034699864580637435114, 9.531843796425403344433679608492, 10.57689597773280735553669507498, 11.97413054261582370033651680060, 13.56339706571102621655797947069, 15.96849769400455349787404246440, 17.02950834184976504205130192613, 17.73610505753424173474109583908

Graph of the ZZ-function along the critical line