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Properties

Label 2-1759-1759.1758-c0-0-7
Degree 22
Conductor 17591759
Sign 11
Analytic cond. 0.8778550.877855
Root an. cond. 0.9369390.936939
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.53·5-s + 8-s + 9-s − 1.53·10-s + 1.53·11-s − 1.87·13-s − 16-s + 1.53·17-s − 18-s − 1.53·22-s + 0.347·23-s + 1.34·25-s + 1.87·26-s − 31-s − 1.53·34-s + 1.53·40-s − 1.87·41-s − 1.87·43-s + 1.53·45-s − 0.347·46-s − 1.87·47-s + 49-s − 1.34·50-s + 0.347·53-s + 2.34·55-s + 62-s + ⋯
L(s)  = 1  − 2-s + 1.53·5-s + 8-s + 9-s − 1.53·10-s + 1.53·11-s − 1.87·13-s − 16-s + 1.53·17-s − 18-s − 1.53·22-s + 0.347·23-s + 1.34·25-s + 1.87·26-s − 31-s − 1.53·34-s + 1.53·40-s − 1.87·41-s − 1.87·43-s + 1.53·45-s − 0.347·46-s − 1.87·47-s + 49-s − 1.34·50-s + 0.347·53-s + 2.34·55-s + 62-s + ⋯

Functional equation

Λ(s)=(1759s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1759s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17591759
Sign: 11
Analytic conductor: 0.8778550.877855
Root analytic conductor: 0.9369390.936939
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1759(1758,⋅)\chi_{1759} (1758, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1759, ( :0), 1)(2,\ 1759,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.93026753030.9302675303
L(12)L(\frac12) ≈\approx 0.93026753030.9302675303
L(1)L(1) 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)
bad1759 1−T 1 - T
good2 1+T+T2 1 + T + T^{2}
3 1−T2 1 - T^{2}
5 1−1.53T+T2 1 - 1.53T + T^{2}
7 1−T2 1 - T^{2}
11 1−1.53T+T2 1 - 1.53T + T^{2}
13 1+1.87T+T2 1 + 1.87T + T^{2}
17 1−1.53T+T2 1 - 1.53T + T^{2}
19 1−T2 1 - T^{2}
23 1−0.347T+T2 1 - 0.347T + T^{2}
29 1−T2 1 - T^{2}
31 1+T+T2 1 + T + T^{2}
37 1−T2 1 - T^{2}
41 1+1.87T+T2 1 + 1.87T + T^{2}
43 1+1.87T+T2 1 + 1.87T + T^{2}
47 1+1.87T+T2 1 + 1.87T + T^{2}
53 1−0.347T+T2 1 - 0.347T + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1−0.347T+T2 1 - 0.347T + T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1−2T+T2 1 - 2T + T^{2}
97 1−T2 1 - 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

−9.594561576886732151215276120247, −9.061196552208389652710906885188, −8.007014336265295763649067814375, −7.08214067452355679434743392122, −6.64653186810739069561181595495, −5.33970252119321555311650545523, −4.78912658868857599530870358859, −3.51122826036390459186817033217, −1.95418773147952980826953839856, −1.36010945604830553381911373416, 1.36010945604830553381911373416, 1.95418773147952980826953839856, 3.51122826036390459186817033217, 4.78912658868857599530870358859, 5.33970252119321555311650545523, 6.64653186810739069561181595495, 7.08214067452355679434743392122, 8.007014336265295763649067814375, 9.061196552208389652710906885188, 9.594561576886732151215276120247

Graph of the ZZ-function along the critical line