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Properties

Label 2-315-1.1-c7-0-59
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  − 18·2-s + 196·4-s + 125·5-s + 343·7-s − 1.22e3·8-s − 2.25e3·10-s + 8.01e3·11-s − 1.78e3·13-s − 6.17e3·14-s − 3.05e3·16-s − 8.35e3·17-s − 5.88e3·19-s + 2.45e4·20-s − 1.44e5·22-s + 7.77e4·23-s + 1.56e4·25-s + 3.21e4·26-s + 6.72e4·28-s − 1.55e5·29-s − 3.10e5·31-s + 2.11e5·32-s + 1.50e5·34-s + 4.28e4·35-s − 4.33e5·37-s + 1.05e5·38-s − 1.53e5·40-s − 3.57e5·41-s + ⋯
L(s)  = 1  − 1.59·2-s + 1.53·4-s + 0.447·5-s + 0.377·7-s − 0.845·8-s − 0.711·10-s + 1.81·11-s − 0.225·13-s − 0.601·14-s − 0.186·16-s − 0.412·17-s − 0.196·19-s + 0.684·20-s − 2.88·22-s + 1.33·23-s + 1/5·25-s + 0.358·26-s + 0.578·28-s − 1.18·29-s − 1.86·31-s + 1.14·32-s + 0.656·34-s + 0.169·35-s − 1.40·37-s + 0.313·38-s − 0.377·40-s − 0.811·41-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+9pT+p7T2 1 + 9 p T + p^{7} T^{2}
11 1−8016T+p7T2 1 - 8016 T + p^{7} T^{2}
13 1+1786T+p7T2 1 + 1786 T + p^{7} T^{2}
17 1+8358T+p7T2 1 + 8358 T + p^{7} T^{2}
19 1+5884T+p7T2 1 + 5884 T + p^{7} T^{2}
23 1−77700T+p7T2 1 - 77700 T + p^{7} T^{2}
29 1+155742T+p7T2 1 + 155742 T + p^{7} T^{2}
31 1+10000pT+p7T2 1 + 10000 p T + p^{7} T^{2}
37 1+433618T+p7T2 1 + 433618 T + p^{7} T^{2}
41 1+357942T+p7T2 1 + 357942 T + p^{7} T^{2}
43 1+724492T+p7T2 1 + 724492 T + p^{7} T^{2}
47 1+175320T+p7T2 1 + 175320 T + p^{7} T^{2}
53 1+132198T+p7T2 1 + 132198 T + p^{7} T^{2}
59 1+44892pT+p7T2 1 + 44892 p T + p^{7} T^{2}
61 1−835478T+p7T2 1 - 835478 T + p^{7} T^{2}
67 1−3486308T+p7T2 1 - 3486308 T + p^{7} T^{2}
71 1−2872260T+p7T2 1 - 2872260 T + p^{7} T^{2}
73 1−5951882T+p7T2 1 - 5951882 T + p^{7} T^{2}
79 1+1680904T+p7T2 1 + 1680904 T + p^{7} T^{2}
83 1+3577524T+p7T2 1 + 3577524 T + p^{7} T^{2}
89 1−6254826T+p7T2 1 - 6254826 T + p^{7} T^{2}
97 1+5257054T+p7T2 1 + 5257054 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

−9.617048099366130674806335954011, −9.151661369503961962117148145250, −8.405675446409159131043949560336, −7.13188360919454764319842406104, −6.61636499852554801866918111058, −5.13646523835169714366678375839, −3.64173694804000630406893618966, −1.94298647854702915529021799917, −1.33202535904229649704791322438, 0, 1.33202535904229649704791322438, 1.94298647854702915529021799917, 3.64173694804000630406893618966, 5.13646523835169714366678375839, 6.61636499852554801866918111058, 7.13188360919454764319842406104, 8.405675446409159131043949560336, 9.151661369503961962117148145250, 9.617048099366130674806335954011

Graph of the ZZ-function along the critical line