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Properties

Label 2-155-1.1-c1-0-10
Degree 22
Conductor 155155
Sign −1-1
Analytic cond. 1.237681.23768
Root an. cond. 1.112511.11251
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s − 2·9-s − 4·11-s + 2·12-s − 6·13-s + 15-s + 4·16-s + 5·17-s − 19-s + 2·20-s + 8·23-s + 25-s + 5·27-s − 10·29-s − 31-s + 4·33-s + 4·36-s + 37-s + 6·39-s − 3·41-s − 7·43-s + 8·44-s + 2·45-s − 6·47-s − 4·48-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s − 1.20·11-s + 0.577·12-s − 1.66·13-s + 0.258·15-s + 16-s + 1.21·17-s − 0.229·19-s + 0.447·20-s + 1.66·23-s + 1/5·25-s + 0.962·27-s − 1.85·29-s − 0.179·31-s + 0.696·33-s + 2/3·36-s + 0.164·37-s + 0.960·39-s − 0.468·41-s − 1.06·43-s + 1.20·44-s + 0.298·45-s − 0.875·47-s − 0.577·48-s + ⋯

Functional equation

Λ(s)=(155s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(155s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 155155    =    5⋅315 \cdot 31
Sign: −1-1
Analytic conductor: 1.237681.23768
Root analytic conductor: 1.112511.11251
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 155, ( :1/2), −1)(2,\ 155,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)Isogeny Class over Fp\mathbf{F}_p
bad5 1+T 1 + T
31 1+T 1 + T
good2 1+pT2 1 + p T^{2} 1.2.a
3 1+T+pT2 1 + T + p T^{2} 1.3.b
7 1+pT2 1 + p T^{2} 1.7.a
11 1+4T+pT2 1 + 4 T + p T^{2} 1.11.e
13 1+6T+pT2 1 + 6 T + p T^{2} 1.13.g
17 1−5T+pT2 1 - 5 T + p T^{2} 1.17.af
19 1+T+pT2 1 + T + p T^{2} 1.19.b
23 1−8T+pT2 1 - 8 T + p T^{2} 1.23.ai
29 1+10T+pT2 1 + 10 T + p T^{2} 1.29.k
37 1−T+pT2 1 - T + p T^{2} 1.37.ab
41 1+3T+pT2 1 + 3 T + p T^{2} 1.41.d
43 1+7T+pT2 1 + 7 T + p T^{2} 1.43.h
47 1+6T+pT2 1 + 6 T + p T^{2} 1.47.g
53 1−5T+pT2 1 - 5 T + p T^{2} 1.53.af
59 1−11T+pT2 1 - 11 T + p T^{2} 1.59.al
61 1+12T+pT2 1 + 12 T + p T^{2} 1.61.m
67 1+2T+pT2 1 + 2 T + p T^{2} 1.67.c
71 1−9T+pT2 1 - 9 T + p T^{2} 1.71.aj
73 1+9T+pT2 1 + 9 T + p T^{2} 1.73.j
79 1+10T+pT2 1 + 10 T + p T^{2} 1.79.k
83 1−9T+pT2 1 - 9 T + p T^{2} 1.83.aj
89 1+pT2 1 + p T^{2} 1.89.a
97 1+14T+pT2 1 + 14 T + p T^{2} 1.97.o
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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

−12.53492345415150194745579045042, −11.52172098246188687978252120211, −10.40498491793852039325606374168, −9.458542992010369876640633952126, −8.232098142144935795769541453640, −7.31046477239498505884334187868, −5.44890211749766646861026772927, −4.92691899597547312483394760424, −3.14330660201336049402162930576, 0, 3.14330660201336049402162930576, 4.92691899597547312483394760424, 5.44890211749766646861026772927, 7.31046477239498505884334187868, 8.232098142144935795769541453640, 9.458542992010369876640633952126, 10.40498491793852039325606374168, 11.52172098246188687978252120211, 12.53492345415150194745579045042

Graph of the ZZ-function along the critical line