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Properties

Label 2-115-1.1-c3-0-18
Degree 22
Conductor 115115
Sign −1-1
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
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  + 2·2-s − 3·3-s − 4·4-s + 5·5-s − 6·6-s − 2·7-s − 24·8-s − 18·9-s + 10·10-s − 16·11-s + 12·12-s − 47·13-s − 4·14-s − 15·15-s − 16·16-s − 24·17-s − 36·18-s − 56·19-s − 20·20-s + 6·21-s − 32·22-s − 23·23-s + 72·24-s + 25·25-s − 94·26-s + 135·27-s + 8·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.107·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.438·11-s + 0.288·12-s − 1.00·13-s − 0.0763·14-s − 0.258·15-s − 1/4·16-s − 0.342·17-s − 0.471·18-s − 0.676·19-s − 0.223·20-s + 0.0623·21-s − 0.310·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.709·26-s + 0.962·27-s + 0.0539·28-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −1-1
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 115, ( :3/2), −1)(2,\ 115,\ (\ :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)
bad5 1−pT 1 - p T
23 1+pT 1 + p T
good2 1−pT+p3T2 1 - p T + p^{3} T^{2}
3 1+pT+p3T2 1 + p T + p^{3} T^{2}
7 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
11 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
13 1+47T+p3T2 1 + 47 T + p^{3} T^{2}
17 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
19 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
29 1−85T+p3T2 1 - 85 T + p^{3} T^{2}
31 1−67T+p3T2 1 - 67 T + p^{3} T^{2}
37 1−104T+p3T2 1 - 104 T + p^{3} T^{2}
41 1+53T+p3T2 1 + 53 T + p^{3} T^{2}
43 1+234T+p3T2 1 + 234 T + p^{3} T^{2}
47 1−285T+p3T2 1 - 285 T + p^{3} T^{2}
53 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
59 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
61 1+764T+p3T2 1 + 764 T + p^{3} T^{2}
67 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
71 1+289T+p3T2 1 + 289 T + p^{3} T^{2}
73 1+225T+p3T2 1 + 225 T + p^{3} T^{2}
79 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
83 1−684T+p3T2 1 - 684 T + p^{3} T^{2}
89 1+1370T+p3T2 1 + 1370 T + p^{3} T^{2}
97 1+110T+p3T2 1 + 110 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

−12.62473830138726021822781319226, −11.81738890813518215519086870923, −10.54115165425474789150963672913, −9.438991490602391872746786499377, −8.278925163476720277372220008673, −6.52676103716535718236934152456, −5.50555403756233399211378041138, −4.56036285838337283133747946354, −2.77522407183088203267266037926, 0, 2.77522407183088203267266037926, 4.56036285838337283133747946354, 5.50555403756233399211378041138, 6.52676103716535718236934152456, 8.278925163476720277372220008673, 9.438991490602391872746786499377, 10.54115165425474789150963672913, 11.81738890813518215519086870923, 12.62473830138726021822781319226

Graph of the ZZ-function along the critical line