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Properties

Label 2-175-1.1-c5-0-33
Degree 22
Conductor 175175
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·2-s + 14·3-s + 68·4-s + 140·6-s + 49·7-s + 360·8-s − 47·9-s + 232·11-s + 952·12-s + 140·13-s + 490·14-s + 1.42e3·16-s + 1.72e3·17-s − 470·18-s − 98·19-s + 686·21-s + 2.32e3·22-s − 1.82e3·23-s + 5.04e3·24-s + 1.40e3·26-s − 4.06e3·27-s + 3.33e3·28-s + 3.41e3·29-s − 7.64e3·31-s + 2.72e3·32-s + 3.24e3·33-s + 1.72e4·34-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.898·3-s + 17/8·4-s + 1.58·6-s + 0.377·7-s + 1.98·8-s − 0.193·9-s + 0.578·11-s + 1.90·12-s + 0.229·13-s + 0.668·14-s + 1.39·16-s + 1.44·17-s − 0.341·18-s − 0.0622·19-s + 0.339·21-s + 1.02·22-s − 0.718·23-s + 1.78·24-s + 0.406·26-s − 1.07·27-s + 0.803·28-s + 0.754·29-s − 1.42·31-s + 0.469·32-s + 0.519·33-s + 2.55·34-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: 11
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: 00
Selberg data: (2, 175, ( :5/2), 1)(2,\ 175,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 7.8617419257.861741925
L(12)L(\frac12) ≈\approx 7.8617419257.861741925
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 1−5pT+p5T2 1 - 5 p T + p^{5} T^{2}
3 1−14T+p5T2 1 - 14 T + p^{5} T^{2}
11 1−232T+p5T2 1 - 232 T + p^{5} T^{2}
13 1−140T+p5T2 1 - 140 T + p^{5} T^{2}
17 1−1722T+p5T2 1 - 1722 T + p^{5} T^{2}
19 1+98T+p5T2 1 + 98 T + p^{5} T^{2}
23 1+1824T+p5T2 1 + 1824 T + p^{5} T^{2}
29 1−3418T+p5T2 1 - 3418 T + p^{5} T^{2}
31 1+7644T+p5T2 1 + 7644 T + p^{5} T^{2}
37 1−10398T+p5T2 1 - 10398 T + p^{5} T^{2}
41 1+17962T+p5T2 1 + 17962 T + p^{5} T^{2}
43 1+10880T+p5T2 1 + 10880 T + p^{5} T^{2}
47 1+9324T+p5T2 1 + 9324 T + p^{5} T^{2}
53 1+2262T+p5T2 1 + 2262 T + p^{5} T^{2}
59 1+2730T+p5T2 1 + 2730 T + p^{5} T^{2}
61 1−25648T+p5T2 1 - 25648 T + p^{5} T^{2}
67 1−48404T+p5T2 1 - 48404 T + p^{5} T^{2}
71 1+58560T+p5T2 1 + 58560 T + p^{5} T^{2}
73 1+68082T+p5T2 1 + 68082 T + p^{5} T^{2}
79 1−31784T+p5T2 1 - 31784 T + p^{5} T^{2}
83 1−20538T+p5T2 1 - 20538 T + p^{5} T^{2}
89 1+50582T+p5T2 1 + 50582 T + p^{5} T^{2}
97 1−58506T+p5T2 1 - 58506 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

−12.00612503520481003014390500628, −11.34705271128151429025700355692, −9.959815623940900851218828111402, −8.539979483506156752982280547873, −7.48808886275778372411031332056, −6.20397807893814053100474613341, −5.19817163096014321019995727184, −3.88553469698264429985104906666, −3.09069771676693723400149567648, −1.77211365157158500631676652077, 1.77211365157158500631676652077, 3.09069771676693723400149567648, 3.88553469698264429985104906666, 5.19817163096014321019995727184, 6.20397807893814053100474613341, 7.48808886275778372411031332056, 8.539979483506156752982280547873, 9.959815623940900851218828111402, 11.34705271128151429025700355692, 12.00612503520481003014390500628

Graph of the ZZ-function along the critical line