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Properties

Label 2-175-1.1-c5-0-45
Degree 22
Conductor 175175
Sign −1-1
Analytic cond. 28.067128.0671
Root an. cond. 5.297845.29784
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 3-s + 32·4-s − 8·6-s − 49·7-s − 242·9-s − 453·11-s − 32·12-s + 969·13-s − 392·14-s − 1.02e3·16-s − 1.63e3·17-s − 1.93e3·18-s − 1.55e3·19-s + 49·21-s − 3.62e3·22-s + 1.65e3·23-s + 7.75e3·26-s + 485·27-s − 1.56e3·28-s − 4.98e3·29-s + 1.19e3·31-s − 8.19e3·32-s + 453·33-s − 1.30e4·34-s − 7.74e3·36-s + 1.10e4·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.0641·3-s + 4-s − 0.0907·6-s − 0.377·7-s − 0.995·9-s − 1.12·11-s − 0.0641·12-s + 1.59·13-s − 0.534·14-s − 16-s − 1.37·17-s − 1.40·18-s − 0.985·19-s + 0.0242·21-s − 1.59·22-s + 0.651·23-s + 2.24·26-s + 0.128·27-s − 0.377·28-s − 1.10·29-s + 0.222·31-s − 1.41·32-s + 0.0724·33-s − 1.94·34-s − 0.995·36-s + 1.32·37-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −1-1
Analytic conductor: 28.067128.0671
Root analytic conductor: 5.297845.29784
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 175, ( :5/2), −1)(2,\ 175,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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+p2T 1 + p^{2} T
good2 1−p3T+p5T2 1 - p^{3} T + p^{5} T^{2}
3 1+T+p5T2 1 + T + p^{5} T^{2}
11 1+453T+p5T2 1 + 453 T + p^{5} T^{2}
13 1−969T+p5T2 1 - 969 T + p^{5} T^{2}
17 1+1637T+p5T2 1 + 1637 T + p^{5} T^{2}
19 1+1550T+p5T2 1 + 1550 T + p^{5} T^{2}
23 1−1654T+p5T2 1 - 1654 T + p^{5} T^{2}
29 1+4985T+p5T2 1 + 4985 T + p^{5} T^{2}
31 1−1192T+p5T2 1 - 1192 T + p^{5} T^{2}
37 1−11018T+p5T2 1 - 11018 T + p^{5} T^{2}
41 1+1728T+p5T2 1 + 1728 T + p^{5} T^{2}
43 1−10814T+p5T2 1 - 10814 T + p^{5} T^{2}
47 1+26237T+p5T2 1 + 26237 T + p^{5} T^{2}
53 1+25936T+p5T2 1 + 25936 T + p^{5} T^{2}
59 1+4580T+p5T2 1 + 4580 T + p^{5} T^{2}
61 1+12488T+p5T2 1 + 12488 T + p^{5} T^{2}
67 1−15848T+p5T2 1 - 15848 T + p^{5} T^{2}
71 1−51792T+p5T2 1 - 51792 T + p^{5} T^{2}
73 1+4846T+p5T2 1 + 4846 T + p^{5} T^{2}
79 1−62765T+p5T2 1 - 62765 T + p^{5} T^{2}
83 1−23644T+p5T2 1 - 23644 T + p^{5} T^{2}
89 1+147300T+p5T2 1 + 147300 T + p^{5} T^{2}
97 1−8343T+p5T2 1 - 8343 T + p^{5} 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

−11.22762224041800675228809909735, −10.99828358343528042887008179455, −9.181167874331142193605915376323, −8.222676307777630185099448288939, −6.52213151916737151304308337614, −5.86413198511598903494692240517, −4.71444307656059463435381149648, −3.50399969940603835112510009698, −2.42266345016135568338486268314, 0, 2.42266345016135568338486268314, 3.50399969940603835112510009698, 4.71444307656059463435381149648, 5.86413198511598903494692240517, 6.52213151916737151304308337614, 8.222676307777630185099448288939, 9.181167874331142193605915376323, 10.99828358343528042887008179455, 11.22762224041800675228809909735

Graph of the ZZ-function along the critical line