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Properties

Label 2-425-1.1-c1-0-17
Degree 22
Conductor 425425
Sign −1-1
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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 + 3-s − 4-s − 6-s − 7-s + 3·8-s − 2·9-s − 4·11-s − 12-s + 13-s + 14-s − 16-s − 17-s + 2·18-s − 6·19-s − 21-s + 4·22-s + 3·24-s − 26-s − 5·27-s + 28-s − 7·31-s − 5·32-s − 4·33-s + 34-s + 2·36-s + 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.242·17-s + 0.471·18-s − 1.37·19-s − 0.218·21-s + 0.852·22-s + 0.612·24-s − 0.196·26-s − 0.962·27-s + 0.188·28-s − 1.25·31-s − 0.883·32-s − 0.696·33-s + 0.171·34-s + 1/3·36-s + 0.657·37-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −1-1
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 425, ( :1/2), −1)(2,\ 425,\ (\ :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
17 1+T 1 + T
good2 1+T+pT2 1 + T + p T^{2}
3 1−T+pT2 1 - T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+2T+pT2 1 + 2 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+11T+pT2 1 + 11 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+11T+pT2 1 + 11 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−16T+pT2 1 - 16 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

−10.53968803223588427431128368094, −9.695773631162169221062385219331, −8.769729262878591744005790361382, −8.276089527076399479578158616124, −7.36741210347521760907587460286, −5.98997398809149648195348749724, −4.83232908272581324807007495531, −3.55455197944795936707370012277, −2.20815571195466993808646287982, 0, 2.20815571195466993808646287982, 3.55455197944795936707370012277, 4.83232908272581324807007495531, 5.98997398809149648195348749724, 7.36741210347521760907587460286, 8.276089527076399479578158616124, 8.769729262878591744005790361382, 9.695773631162169221062385219331, 10.53968803223588427431128368094

Graph of the ZZ-function along the critical line