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Properties

Label 2-825-1.1-c1-0-27
Degree 22
Conductor 825825
Sign −1-1
Analytic cond. 6.587656.58765
Root an. cond. 2.566642.56664
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 − 7-s + 9-s − 11-s − 2·12-s − 13-s + 4·16-s − 6·17-s − 7·19-s − 21-s + 6·23-s + 27-s + 2·28-s − 6·29-s − 7·31-s − 33-s − 2·36-s + 2·37-s − 39-s − 6·41-s − 43-s + 2·44-s + 4·48-s − 6·49-s − 6·51-s + 2·52-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 1.45·17-s − 1.60·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.25·31-s − 0.174·33-s − 1/3·36-s + 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.152·43-s + 0.301·44-s + 0.577·48-s − 6/7·49-s − 0.840·51-s + 0.277·52-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −1-1
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 825, ( :1/2), −1)(2,\ 825,\ (\ :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)
bad3 1−T 1 - T
5 1 1
11 1+T 1 + T
good2 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+7T+pT2 1 + 7 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−17T+pT2 1 - 17 T + p 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

−9.567309717603522064679693310635, −8.955732103392000591141257065944, −8.367781725670412632475845953636, −7.30708842123598942599324121736, −6.37594318082116078700438020966, −5.13071370263278861735871300481, −4.31466281424388077222299797352, −3.35971135428784856363070844882, −2.04859260711079521330868103173, 0, 2.04859260711079521330868103173, 3.35971135428784856363070844882, 4.31466281424388077222299797352, 5.13071370263278861735871300481, 6.37594318082116078700438020966, 7.30708842123598942599324121736, 8.367781725670412632475845953636, 8.955732103392000591141257065944, 9.567309717603522064679693310635

Graph of the ZZ-function along the critical line