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Properties

Label 2-95e2-1.1-c1-0-484
Degree 22
Conductor 90259025
Sign −1-1
Analytic cond. 72.064972.0649
Root an. cond. 8.489108.48910
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  + 2·2-s + 2·4-s + 4·7-s − 3·9-s − 11-s − 2·13-s + 8·14-s − 4·16-s + 2·17-s − 6·18-s − 2·22-s − 6·23-s − 4·26-s + 8·28-s − 9·29-s + 7·31-s − 8·32-s + 4·34-s − 6·36-s + 2·37-s − 2·41-s − 2·43-s − 2·44-s − 12·46-s − 6·47-s + 9·49-s − 4·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.51·7-s − 9-s − 0.301·11-s − 0.554·13-s + 2.13·14-s − 16-s + 0.485·17-s − 1.41·18-s − 0.426·22-s − 1.25·23-s − 0.784·26-s + 1.51·28-s − 1.67·29-s + 1.25·31-s − 1.41·32-s + 0.685·34-s − 36-s + 0.328·37-s − 0.312·41-s − 0.304·43-s − 0.301·44-s − 1.76·46-s − 0.875·47-s + 9/7·49-s − 0.554·52-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 72.064972.0649
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9025, ( :1/2), −1)(2,\ 9025,\ (\ :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)
bad5 1 1
19 1 1
good2 1−pT+pT2 1 - p T + p T^{2}
3 1+pT2 1 + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+9T+pT2 1 + 9 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+2T+pT2 1 + 2 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+T+pT2 1 + T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−11T+pT2 1 - 11 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−7.49108759469514968488634151108, −6.32140657880002311696717149039, −5.83425220431077261197640060450, −5.16878200823707144795848625245, −4.73393782803949320805165463406, −4.00846770547569728687671158073, −3.16443892797854562205339038031, −2.41426315202176417098684876832, −1.64347015071941685368598744615, 0, 1.64347015071941685368598744615, 2.41426315202176417098684876832, 3.16443892797854562205339038031, 4.00846770547569728687671158073, 4.73393782803949320805165463406, 5.16878200823707144795848625245, 5.83425220431077261197640060450, 6.32140657880002311696717149039, 7.49108759469514968488634151108

Graph of the ZZ-function along the critical line