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Properties

Label 2-315-1.1-c3-0-15
Degree 22
Conductor 315315
Sign −1-1
Analytic cond. 18.585618.5856
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4-s − 5·5-s + 7·7-s + 21·8-s + 15·10-s − 60·11-s + 38·13-s − 21·14-s − 71·16-s + 84·17-s + 110·19-s − 5·20-s + 180·22-s − 120·23-s + 25·25-s − 114·26-s + 7·28-s − 162·29-s + 236·31-s + 45·32-s − 252·34-s − 35·35-s − 376·37-s − 330·38-s − 105·40-s + 126·41-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s − 0.447·5-s + 0.377·7-s + 0.928·8-s + 0.474·10-s − 1.64·11-s + 0.810·13-s − 0.400·14-s − 1.10·16-s + 1.19·17-s + 1.32·19-s − 0.0559·20-s + 1.74·22-s − 1.08·23-s + 1/5·25-s − 0.859·26-s + 0.0472·28-s − 1.03·29-s + 1.36·31-s + 0.248·32-s − 1.27·34-s − 0.169·35-s − 1.67·37-s − 1.40·38-s − 0.415·40-s + 0.479·41-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 18.585618.5856
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 315, ( :3/2), −1)(2,\ 315,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
7 1−pT 1 - p T
good2 1+3T+p3T2 1 + 3 T + p^{3} T^{2}
11 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
13 1−38T+p3T2 1 - 38 T + p^{3} T^{2}
17 1−84T+p3T2 1 - 84 T + p^{3} T^{2}
19 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
23 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
29 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
31 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
37 1+376T+p3T2 1 + 376 T + p^{3} T^{2}
41 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
43 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
47 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
53 1+582T+p3T2 1 + 582 T + p^{3} T^{2}
59 1+492T+p3T2 1 + 492 T + p^{3} T^{2}
61 1+880T+p3T2 1 + 880 T + p^{3} T^{2}
67 1+826T+p3T2 1 + 826 T + p^{3} T^{2}
71 1−666T+p3T2 1 - 666 T + p^{3} T^{2}
73 1+826T+p3T2 1 + 826 T + p^{3} T^{2}
79 1+592T+p3T2 1 + 592 T + p^{3} T^{2}
83 1+792T+p3T2 1 + 792 T + p^{3} T^{2}
89 1+1002T+p3T2 1 + 1002 T + p^{3} T^{2}
97 1−1442T+p3T2 1 - 1442 T + p^{3} 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

−10.50330158489365955882557188522, −9.911381883658121032798486635180, −8.767931103556554796481045612733, −7.81628435365882593277923531714, −7.57444477318687232989936227330, −5.75829389662466937868867493509, −4.70691641096985925218101864826, −3.22395287623718505312847664315, −1.44446039907941357054892954820, 0, 1.44446039907941357054892954820, 3.22395287623718505312847664315, 4.70691641096985925218101864826, 5.75829389662466937868867493509, 7.57444477318687232989936227330, 7.81628435365882593277923531714, 8.767931103556554796481045612733, 9.911381883658121032798486635180, 10.50330158489365955882557188522

Graph of the ZZ-function along the critical line