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Properties

Label 2-175-1.1-c9-0-59
Degree 22
Conductor 175175
Sign −1-1
Analytic cond. 90.131290.1312
Root an. cond. 9.493749.49374
Motivic weight 99
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28·2-s + 116·3-s + 272·4-s − 3.24e3·6-s − 2.40e3·7-s + 6.72e3·8-s − 6.22e3·9-s − 2.55e4·11-s + 3.15e4·12-s + 4.23e4·13-s + 6.72e4·14-s − 3.27e5·16-s + 5.26e5·17-s + 1.74e5·18-s − 3.50e5·19-s − 2.78e5·21-s + 7.15e5·22-s + 6.21e5·23-s + 7.79e5·24-s − 1.18e6·26-s − 3.00e6·27-s − 6.53e5·28-s + 6.72e6·29-s − 6.41e6·31-s + 5.72e6·32-s − 2.96e6·33-s − 1.47e7·34-s + ⋯
L(s)  = 1  − 1.23·2-s + 0.826·3-s + 0.531·4-s − 1.02·6-s − 0.377·7-s + 0.580·8-s − 0.316·9-s − 0.526·11-s + 0.439·12-s + 0.410·13-s + 0.467·14-s − 1.24·16-s + 1.52·17-s + 0.391·18-s − 0.616·19-s − 0.312·21-s + 0.651·22-s + 0.463·23-s + 0.479·24-s − 0.508·26-s − 1.08·27-s − 0.200·28-s + 1.76·29-s − 1.24·31-s + 0.965·32-s − 0.435·33-s − 1.89·34-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −1-1
Analytic conductor: 90.131290.1312
Root analytic conductor: 9.493749.49374
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 175, ( :9/2), −1)(2,\ 175,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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
7 1+p4T 1 + p^{4} T
good2 1+7p2T+p9T2 1 + 7 p^{2} T + p^{9} T^{2}
3 1−116T+p9T2 1 - 116 T + p^{9} T^{2}
11 1+25548T+p9T2 1 + 25548 T + p^{9} T^{2}
13 1−42306T+p9T2 1 - 42306 T + p^{9} T^{2}
17 1−526342T+p9T2 1 - 526342 T + p^{9} T^{2}
19 1+350060T+p9T2 1 + 350060 T + p^{9} T^{2}
23 1−621976T+p9T2 1 - 621976 T + p^{9} T^{2}
29 1−6720430T+p9T2 1 - 6720430 T + p^{9} T^{2}
31 1+6412208T+p9T2 1 + 6412208 T + p^{9} T^{2}
37 1−2317682T+p9T2 1 - 2317682 T + p^{9} T^{2}
41 1+10224678T+p9T2 1 + 10224678 T + p^{9} T^{2}
43 1+30114004T+p9T2 1 + 30114004 T + p^{9} T^{2}
47 1−23644912T+p9T2 1 - 23644912 T + p^{9} T^{2}
53 1+57292654T+p9T2 1 + 57292654 T + p^{9} T^{2}
59 1−84934780T+p9T2 1 - 84934780 T + p^{9} T^{2}
61 1−14677822T+p9T2 1 - 14677822 T + p^{9} T^{2}
67 1−244557812T+p9T2 1 - 244557812 T + p^{9} T^{2}
71 1−61901952T+p9T2 1 - 61901952 T + p^{9} T^{2}
73 1−283763726T+p9T2 1 - 283763726 T + p^{9} T^{2}
79 1−276107480T+p9T2 1 - 276107480 T + p^{9} T^{2}
83 1−72995956T+p9T2 1 - 72995956 T + p^{9} T^{2}
89 1+896368470T+p9T2 1 + 896368470 T + p^{9} T^{2}
97 1+1205809578T+p9T2 1 + 1205809578 T + p^{9} 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

−10.19112103600048002666938633816, −9.435712170282542004710924262217, −8.439744059232341352264003846412, −7.977719110662855327091087384402, −6.76732199824049326380432917954, −5.27286289517867541834394924791, −3.63840962846763431560313798285, −2.52153495373356647184053484745, −1.21349497488542371736328244652, 0, 1.21349497488542371736328244652, 2.52153495373356647184053484745, 3.63840962846763431560313798285, 5.27286289517867541834394924791, 6.76732199824049326380432917954, 7.977719110662855327091087384402, 8.439744059232341352264003846412, 9.435712170282542004710924262217, 10.19112103600048002666938633816

Graph of the ZZ-function along the critical line