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Properties

Label 2-95e2-1.1-c1-0-212
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-s − 4-s − 2·7-s + 3·8-s − 3·9-s − 4·11-s − 2·13-s + 2·14-s − 16-s − 4·17-s + 3·18-s + 4·22-s + 6·23-s + 2·26-s + 2·28-s + 6·29-s + 4·31-s − 5·32-s + 4·34-s + 3·36-s − 10·37-s + 10·41-s − 2·43-s + 4·44-s − 6·46-s + 6·47-s − 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 9-s − 1.20·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.852·22-s + 1.25·23-s + 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.685·34-s + 1/2·36-s − 1.64·37-s + 1.56·41-s − 0.304·43-s + 0.603·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-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+T+pT2 1 + T + p T^{2}
3 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 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 1−4T+pT2 1 - 4 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−10T+pT2 1 - 10 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 1−10T+pT2 1 - 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−18T+pT2 1 - 18 T + p T^{2}
89 1−2T+pT2 1 - 2 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

−7.50136799669702123180560377799, −6.87231635746065961134156986184, −6.07990432071003204598796847667, −5.15049393566020071335799991802, −4.84132098018455846662836521172, −3.78973024595814975369272522800, −2.84788762764959730109880934230, −2.32358969255822247087366120152, −0.816254076128081240605604169498, 0, 0.816254076128081240605604169498, 2.32358969255822247087366120152, 2.84788762764959730109880934230, 3.78973024595814975369272522800, 4.84132098018455846662836521172, 5.15049393566020071335799991802, 6.07990432071003204598796847667, 6.87231635746065961134156986184, 7.50136799669702123180560377799

Graph of the ZZ-function along the critical line