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Properties

Label 2-95e2-1.1-c1-0-471
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  + 2·3-s − 2·4-s + 4·7-s + 9-s + 3·11-s − 4·12-s − 2·13-s + 4·16-s − 6·17-s + 8·21-s − 4·27-s − 8·28-s − 3·29-s − 7·31-s + 6·33-s − 2·36-s − 8·37-s − 4·39-s − 6·41-s + 4·43-s − 6·44-s − 6·47-s + 8·48-s + 9·49-s − 12·51-s + 4·52-s + 6·53-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1.51·7-s + 1/3·9-s + 0.904·11-s − 1.15·12-s − 0.554·13-s + 16-s − 1.45·17-s + 1.74·21-s − 0.769·27-s − 1.51·28-s − 0.557·29-s − 1.25·31-s + 1.04·33-s − 1/3·36-s − 1.31·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.904·44-s − 0.875·47-s + 1.15·48-s + 9/7·49-s − 1.68·51-s + 0.554·52-s + 0.824·53-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+pT2 1 + p T^{2}
3 1−2T+pT2 1 - 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+15T+pT2 1 + 15 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−7.54077871130760231857953639957, −7.02670731199975696065066202804, −5.86857745605201058059579690124, −5.10667495915163109837412560658, −4.48119941130551282907573401981, −3.93584614361416858298673264501, −3.15451099160830095639023268055, −2.02950889034593554805831885097, −1.55378852660460568540896216770, 0, 1.55378852660460568540896216770, 2.02950889034593554805831885097, 3.15451099160830095639023268055, 3.93584614361416858298673264501, 4.48119941130551282907573401981, 5.10667495915163109837412560658, 5.86857745605201058059579690124, 7.02670731199975696065066202804, 7.54077871130760231857953639957

Graph of the ZZ-function along the critical line