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Properties

Label 2-175-7.6-c4-0-33
Degree 22
Conductor 175175
Sign 11
Analytic cond. 18.089718.0897
Root an. cond. 4.253204.25320
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.37·2-s + 24.6·4-s + 49·7-s + 54.9·8-s + 81·9-s − 6.98·11-s + 312.·14-s − 43.5·16-s + 516.·18-s − 44.5·22-s + 1.02e3·23-s + 1.20e3·28-s + 372.·29-s − 1.15e3·32-s + 1.99e3·36-s − 1.44e3·37-s − 3.02e3·43-s − 171.·44-s + 6.54e3·46-s + 2.40e3·49-s − 5.58e3·53-s + 2.69e3·56-s + 2.37e3·58-s + 3.96e3·63-s − 6.68e3·64-s − 8.96e3·67-s − 9.81e3·71-s + ⋯
L(s)  = 1  + 1.59·2-s + 1.53·4-s + 0.999·7-s + 0.859·8-s + 9-s − 0.0577·11-s + 1.59·14-s − 0.170·16-s + 1.59·18-s − 0.0919·22-s + 1.94·23-s + 1.53·28-s + 0.443·29-s − 1.13·32-s + 1.53·36-s − 1.05·37-s − 1.63·43-s − 0.0888·44-s + 3.09·46-s + 49-s − 1.98·53-s + 0.859·56-s + 0.706·58-s + 63-s − 1.63·64-s − 1.99·67-s − 1.94·71-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 18.089718.0897
Root analytic conductor: 4.253204.25320
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ175(76,⋅)\chi_{175} (76, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :2), 1)(2,\ 175,\ (\ :2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 5.1685707845.168570784
L(12)L(\frac12) ≈\approx 5.1685707845.168570784
L(3)L(3) 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 1−49T 1 - 49T
good2 1−6.37T+16T2 1 - 6.37T + 16T^{2}
3 1−81T2 1 - 81T^{2}
11 1+6.98T+1.46e4T2 1 + 6.98T + 1.46e4T^{2}
13 1−2.85e4T2 1 - 2.85e4T^{2}
17 1−8.35e4T2 1 - 8.35e4T^{2}
19 1−1.30e5T2 1 - 1.30e5T^{2}
23 1−1.02e3T+2.79e5T2 1 - 1.02e3T + 2.79e5T^{2}
29 1−372.T+7.07e5T2 1 - 372.T + 7.07e5T^{2}
31 1−9.23e5T2 1 - 9.23e5T^{2}
37 1+1.44e3T+1.87e6T2 1 + 1.44e3T + 1.87e6T^{2}
41 1−2.82e6T2 1 - 2.82e6T^{2}
43 1+3.02e3T+3.41e6T2 1 + 3.02e3T + 3.41e6T^{2}
47 1−4.87e6T2 1 - 4.87e6T^{2}
53 1+5.58e3T+7.89e6T2 1 + 5.58e3T + 7.89e6T^{2}
59 1−1.21e7T2 1 - 1.21e7T^{2}
61 1−1.38e7T2 1 - 1.38e7T^{2}
67 1+8.96e3T+2.01e7T2 1 + 8.96e3T + 2.01e7T^{2}
71 1+9.81e3T+2.54e7T2 1 + 9.81e3T + 2.54e7T^{2}
73 1−2.83e7T2 1 - 2.83e7T^{2}
79 1−1.21e4T+3.89e7T2 1 - 1.21e4T + 3.89e7T^{2}
83 1−4.74e7T2 1 - 4.74e7T^{2}
89 1−6.27e7T2 1 - 6.27e7T^{2}
97 1−8.85e7T2 1 - 8.85e7T^{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

−12.25686407516953938470389005056, −11.36042688538279057850055176477, −10.45169171331194748284752319263, −8.934189975192986807580554255720, −7.50328206096375990157342518413, −6.55530917650553745204314269291, −5.13417265281011461558069404304, −4.54185732078532777915831013045, −3.19501563042873379804720795473, −1.61020334298225034280974882349, 1.61020334298225034280974882349, 3.19501563042873379804720795473, 4.54185732078532777915831013045, 5.13417265281011461558069404304, 6.55530917650553745204314269291, 7.50328206096375990157342518413, 8.934189975192986807580554255720, 10.45169171331194748284752319263, 11.36042688538279057850055176477, 12.25686407516953938470389005056

Graph of the ZZ-function along the critical line