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Properties

Label 2-175-1.1-c3-0-3
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  − 1.67·2-s − 2.49·3-s − 5.19·4-s + 4.17·6-s + 7·7-s + 22.1·8-s − 20.7·9-s − 57.5·11-s + 12.9·12-s − 45.5·13-s − 11.7·14-s + 4.52·16-s + 92.0·17-s + 34.8·18-s + 125.·19-s − 17.4·21-s + 96.4·22-s + 158.·23-s − 55.1·24-s + 76.2·26-s + 119.·27-s − 36.3·28-s − 40.1·29-s + 49.5·31-s − 184.·32-s + 143.·33-s − 154.·34-s + ⋯
L(s)  = 1  − 0.592·2-s − 0.479·3-s − 0.649·4-s + 0.284·6-s + 0.377·7-s + 0.976·8-s − 0.769·9-s − 1.57·11-s + 0.311·12-s − 0.971·13-s − 0.223·14-s + 0.0707·16-s + 1.31·17-s + 0.455·18-s + 1.51·19-s − 0.181·21-s + 0.934·22-s + 1.43·23-s − 0.468·24-s + 0.575·26-s + 0.849·27-s − 0.245·28-s − 0.257·29-s + 0.287·31-s − 1.01·32-s + 0.757·33-s − 0.777·34-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 0.69700781660.6970078166
L(12)L(\frac12) ≈\approx 0.69700781660.6970078166
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+1.67T+8T2 1 + 1.67T + 8T^{2}
3 1+2.49T+27T2 1 + 2.49T + 27T^{2}
11 1+57.5T+1.33e3T2 1 + 57.5T + 1.33e3T^{2}
13 1+45.5T+2.19e3T2 1 + 45.5T + 2.19e3T^{2}
17 1−92.0T+4.91e3T2 1 - 92.0T + 4.91e3T^{2}
19 1−125.T+6.85e3T2 1 - 125.T + 6.85e3T^{2}
23 1−158.T+1.21e4T2 1 - 158.T + 1.21e4T^{2}
29 1+40.1T+2.43e4T2 1 + 40.1T + 2.43e4T^{2}
31 1−49.5T+2.97e4T2 1 - 49.5T + 2.97e4T^{2}
37 1−231.T+5.06e4T2 1 - 231.T + 5.06e4T^{2}
41 1−169.T+6.89e4T2 1 - 169.T + 6.89e4T^{2}
43 1+147.T+7.95e4T2 1 + 147.T + 7.95e4T^{2}
47 1+67.0T+1.03e5T2 1 + 67.0T + 1.03e5T^{2}
53 1−268.T+1.48e5T2 1 - 268.T + 1.48e5T^{2}
59 1+240.T+2.05e5T2 1 + 240.T + 2.05e5T^{2}
61 1−90.4T+2.26e5T2 1 - 90.4T + 2.26e5T^{2}
67 1+406.T+3.00e5T2 1 + 406.T + 3.00e5T^{2}
71 1−330.T+3.57e5T2 1 - 330.T + 3.57e5T^{2}
73 1−546.T+3.89e5T2 1 - 546.T + 3.89e5T^{2}
79 1+25.3T+4.93e5T2 1 + 25.3T + 4.93e5T^{2}
83 1−376.T+5.71e5T2 1 - 376.T + 5.71e5T^{2}
89 1−1.02e3T+7.04e5T2 1 - 1.02e3T + 7.04e5T^{2}
97 1+942.T+9.12e5T2 1 + 942.T + 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

−12.17641640107063606574338758742, −11.09814476791592664877734107500, −10.18578696555726564955618686112, −9.327452952554630732196799280848, −8.054894951436467442285571365521, −7.46036289748934406731674258742, −5.44584021491038244500703374040, −4.98708788747324604197304626313, −2.94453780951520378804401365235, −0.73552160158185913651901572540, 0.73552160158185913651901572540, 2.94453780951520378804401365235, 4.98708788747324604197304626313, 5.44584021491038244500703374040, 7.46036289748934406731674258742, 8.054894951436467442285571365521, 9.327452952554630732196799280848, 10.18578696555726564955618686112, 11.09814476791592664877734107500, 12.17641640107063606574338758742

Graph of the ZZ-function along the critical line