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Properties

Label 2-475-1.1-c1-0-26
Degree 22
Conductor 475475
Sign −1-1
Analytic cond. 3.792893.79289
Root an. cond. 1.947531.94753
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 + 19-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 + 38-s − 10·41-s − 2·43-s + 4·44-s + 6·46-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 + 0.229·19-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 + 0.162·38-s − 1.56·41-s − 0.304·43-s + 0.603·44-s + 0.884·46-s + ⋯

Functional equation

Λ(s)=(475s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(475s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 475475    =    52⋅195^{2} \cdot 19
Sign: −1-1
Analytic conductor: 3.792893.79289
Root analytic conductor: 1.947531.94753
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 475, ( :1/2), −1)(2,\ 475,\ (\ :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−T 1 - T
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

−10.75372555761590942712128571003, −9.497197103514413153205439327666, −8.884496394435335630734933815790, −7.910634270974764100222562199420, −6.56197008088421338178147885329, −5.66703630365227380430433843993, −4.88960236344040300877313120060, −3.56253018952603651968369317882, −2.71685356445346523746983648243, 0, 2.71685356445346523746983648243, 3.56253018952603651968369317882, 4.88960236344040300877313120060, 5.66703630365227380430433843993, 6.56197008088421338178147885329, 7.910634270974764100222562199420, 8.884496394435335630734933815790, 9.497197103514413153205439327666, 10.75372555761590942712128571003

Graph of the ZZ-function along the critical line