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Properties

Label 2-315-1.1-c7-0-56
Degree 22
Conductor 315315
Sign −1-1
Analytic cond. 98.401298.4012
Root an. cond. 9.919749.91974
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 124·4-s + 125·5-s + 343·7-s − 504·8-s + 250·10-s − 2.72e3·11-s + 2.87e3·13-s + 686·14-s + 1.48e4·16-s − 9.27e3·17-s − 4.30e3·19-s − 1.55e4·20-s − 5.44e3·22-s + 4.15e4·23-s + 1.56e4·25-s + 5.74e3·26-s − 4.25e4·28-s + 3.54e4·29-s − 5.29e4·31-s + 9.42e4·32-s − 1.85e4·34-s + 4.28e4·35-s − 8.40e4·37-s − 8.60e3·38-s − 6.30e4·40-s − 1.80e5·41-s + ⋯
L(s)  = 1  + 0.176·2-s − 0.968·4-s + 0.447·5-s + 0.377·7-s − 0.348·8-s + 0.0790·10-s − 0.617·11-s + 0.362·13-s + 0.0668·14-s + 0.907·16-s − 0.458·17-s − 0.143·19-s − 0.433·20-s − 0.109·22-s + 0.711·23-s + 1/5·25-s + 0.0641·26-s − 0.366·28-s + 0.270·29-s − 0.319·31-s + 0.508·32-s − 0.0809·34-s + 0.169·35-s − 0.272·37-s − 0.0254·38-s − 0.155·40-s − 0.408·41-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 98.401298.4012
Root analytic conductor: 9.919749.91974
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 315, ( :7/2), −1)(2,\ 315,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5 1−p3T 1 - p^{3} T
7 1−p3T 1 - p^{3} T
good2 1−pT+p7T2 1 - p T + p^{7} T^{2}
11 1+2724T+p7T2 1 + 2724 T + p^{7} T^{2}
13 1−2874T+p7T2 1 - 2874 T + p^{7} T^{2}
17 1+9278T+p7T2 1 + 9278 T + p^{7} T^{2}
19 1+4304T+p7T2 1 + 4304 T + p^{7} T^{2}
23 1−41500T+p7T2 1 - 41500 T + p^{7} T^{2}
29 1−35498T+p7T2 1 - 35498 T + p^{7} T^{2}
31 1+52940T+p7T2 1 + 52940 T + p^{7} T^{2}
37 1+84098T+p7T2 1 + 84098 T + p^{7} T^{2}
41 1+180342T+p7T2 1 + 180342 T + p^{7} T^{2}
43 1+33452T+p7T2 1 + 33452 T + p^{7} T^{2}
47 1−136120T+p7T2 1 - 136120 T + p^{7} T^{2}
53 1−23974pT+p7T2 1 - 23974 p T + p^{7} T^{2}
59 1−1553252T+p7T2 1 - 1553252 T + p^{7} T^{2}
61 1−213598T+p7T2 1 - 213598 T + p^{7} T^{2}
67 1−487228T+p7T2 1 - 487228 T + p^{7} T^{2}
71 1+1086000T+p7T2 1 + 1086000 T + p^{7} T^{2}
73 1+5921978T+p7T2 1 + 5921978 T + p^{7} T^{2}
79 1+5429824T+p7T2 1 + 5429824 T + p^{7} T^{2}
83 1+6933404T+p7T2 1 + 6933404 T + p^{7} T^{2}
89 1+262614T+p7T2 1 + 262614 T + p^{7} T^{2}
97 1+522234T+p7T2 1 + 522234 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

−9.978714357961983819834016362634, −8.941675942387986477726587270182, −8.324389533298515320035954047280, −7.08617885469246626824000488682, −5.78613140817393934315282691278, −5.01104748816947260210337486803, −4.00425285494690616056593847642, −2.71048883429078753857118637643, −1.28177727953048994887417205121, 0, 1.28177727953048994887417205121, 2.71048883429078753857118637643, 4.00425285494690616056593847642, 5.01104748816947260210337486803, 5.78613140817393934315282691278, 7.08617885469246626824000488682, 8.324389533298515320035954047280, 8.941675942387986477726587270182, 9.978714357961983819834016362634

Graph of the ZZ-function along the critical line