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Properties

Label 2-175-1.1-c3-0-24
Degree 22
Conductor 175175
Sign 11
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.62·2-s + 8.38·3-s + 13.3·4-s + 38.7·6-s − 7·7-s + 24.9·8-s + 43.3·9-s − 30.1·11-s + 112.·12-s − 88.9·13-s − 32.3·14-s + 8.10·16-s + 4.73·17-s + 200.·18-s + 124.·19-s − 58.7·21-s − 139.·22-s − 20.2·23-s + 208.·24-s − 411.·26-s + 136.·27-s − 93.7·28-s + 134.·29-s − 2.03·31-s − 161.·32-s − 252.·33-s + 21.9·34-s + ⋯
L(s)  = 1  + 1.63·2-s + 1.61·3-s + 1.67·4-s + 2.63·6-s − 0.377·7-s + 1.10·8-s + 1.60·9-s − 0.825·11-s + 2.70·12-s − 1.89·13-s − 0.617·14-s + 0.126·16-s + 0.0675·17-s + 2.62·18-s + 1.50·19-s − 0.610·21-s − 1.34·22-s − 0.183·23-s + 1.77·24-s − 3.10·26-s + 0.976·27-s − 0.632·28-s + 0.858·29-s − 0.0118·31-s − 0.893·32-s − 1.33·33-s + 0.110·34-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 1)(2,\ 175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.9914519775.991451977
L(12)L(\frac12) ≈\approx 5.9914519775.991451977
L(52)L(\frac{5}{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+7T 1 + 7T
good2 1−4.62T+8T2 1 - 4.62T + 8T^{2}
3 1−8.38T+27T2 1 - 8.38T + 27T^{2}
11 1+30.1T+1.33e3T2 1 + 30.1T + 1.33e3T^{2}
13 1+88.9T+2.19e3T2 1 + 88.9T + 2.19e3T^{2}
17 1−4.73T+4.91e3T2 1 - 4.73T + 4.91e3T^{2}
19 1−124.T+6.85e3T2 1 - 124.T + 6.85e3T^{2}
23 1+20.2T+1.21e4T2 1 + 20.2T + 1.21e4T^{2}
29 1−134.T+2.43e4T2 1 - 134.T + 2.43e4T^{2}
31 1+2.03T+2.97e4T2 1 + 2.03T + 2.97e4T^{2}
37 1−141.T+5.06e4T2 1 - 141.T + 5.06e4T^{2}
41 1−95.2T+6.89e4T2 1 - 95.2T + 6.89e4T^{2}
43 1−298.T+7.95e4T2 1 - 298.T + 7.95e4T^{2}
47 1−129.T+1.03e5T2 1 - 129.T + 1.03e5T^{2}
53 1+388.T+1.48e5T2 1 + 388.T + 1.48e5T^{2}
59 1−838.T+2.05e5T2 1 - 838.T + 2.05e5T^{2}
61 1−389.T+2.26e5T2 1 - 389.T + 2.26e5T^{2}
67 1+697.T+3.00e5T2 1 + 697.T + 3.00e5T^{2}
71 1+523.T+3.57e5T2 1 + 523.T + 3.57e5T^{2}
73 1+66.4T+3.89e5T2 1 + 66.4T + 3.89e5T^{2}
79 1+526.T+4.93e5T2 1 + 526.T + 4.93e5T^{2}
83 1+70.0T+5.71e5T2 1 + 70.0T + 5.71e5T^{2}
89 1+9.27T+7.04e5T2 1 + 9.27T + 7.04e5T^{2}
97 1−4.19T+9.12e5T2 1 - 4.19T + 9.12e5T^{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

−12.66747912724749068926902162729, −11.79893074117836167096462282323, −10.15260556784260160675500006575, −9.299588515353147946848677529713, −7.80654746831271593042501031114, −7.09604721812618398512756178251, −5.44703792870315317332294990853, −4.35182692936235663235571516144, −3.03801588927631623923327660640, −2.47221646828391902923246030735, 2.47221646828391902923246030735, 3.03801588927631623923327660640, 4.35182692936235663235571516144, 5.44703792870315317332294990853, 7.09604721812618398512756178251, 7.80654746831271593042501031114, 9.299588515353147946848677529713, 10.15260556784260160675500006575, 11.79893074117836167096462282323, 12.66747912724749068926902162729

Graph of the ZZ-function along the critical line