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Properties

Label 2-1759-1.1-c1-0-112
Degree 22
Conductor 17591759
Sign 11
Analytic cond. 14.045614.0456
Root an. cond. 3.747753.74775
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51·2-s + 0.988·3-s + 4.30·4-s + 0.809·5-s + 2.48·6-s + 2.08·7-s + 5.78·8-s − 2.02·9-s + 2.03·10-s + 0.174·11-s + 4.25·12-s + 1.58·13-s + 5.23·14-s + 0.799·15-s + 5.91·16-s + 2.42·17-s − 5.08·18-s − 0.892·19-s + 3.48·20-s + 2.06·21-s + 0.437·22-s − 4.85·23-s + 5.71·24-s − 4.34·25-s + 3.98·26-s − 4.96·27-s + 8.97·28-s + ⋯
L(s)  = 1  + 1.77·2-s + 0.570·3-s + 2.15·4-s + 0.361·5-s + 1.01·6-s + 0.788·7-s + 2.04·8-s − 0.674·9-s + 0.642·10-s + 0.0525·11-s + 1.22·12-s + 0.440·13-s + 1.40·14-s + 0.206·15-s + 1.47·16-s + 0.587·17-s − 1.19·18-s − 0.204·19-s + 0.778·20-s + 0.449·21-s + 0.0932·22-s − 1.01·23-s + 1.16·24-s − 0.868·25-s + 0.781·26-s − 0.955·27-s + 1.69·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 17591759
Sign: 11
Analytic conductor: 14.045614.0456
Root analytic conductor: 3.747753.74775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1759, ( :1/2), 1)(2,\ 1759,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.6361052696.636105269
L(12)L(\frac12) ≈\approx 6.6361052696.636105269
L(32)L(\frac{3}{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)
bad1759 1−T 1 - T
good2 1−2.51T+2T2 1 - 2.51T + 2T^{2}
3 1−0.988T+3T2 1 - 0.988T + 3T^{2}
5 1−0.809T+5T2 1 - 0.809T + 5T^{2}
7 1−2.08T+7T2 1 - 2.08T + 7T^{2}
11 1−0.174T+11T2 1 - 0.174T + 11T^{2}
13 1−1.58T+13T2 1 - 1.58T + 13T^{2}
17 1−2.42T+17T2 1 - 2.42T + 17T^{2}
19 1+0.892T+19T2 1 + 0.892T + 19T^{2}
23 1+4.85T+23T2 1 + 4.85T + 23T^{2}
29 1−3.57T+29T2 1 - 3.57T + 29T^{2}
31 1+2.95T+31T2 1 + 2.95T + 31T^{2}
37 1−10.4T+37T2 1 - 10.4T + 37T^{2}
41 1−0.384T+41T2 1 - 0.384T + 41T^{2}
43 1+6.89T+43T2 1 + 6.89T + 43T^{2}
47 1+5.84T+47T2 1 + 5.84T + 47T^{2}
53 1−13.1T+53T2 1 - 13.1T + 53T^{2}
59 1+12.1T+59T2 1 + 12.1T + 59T^{2}
61 1−7.96T+61T2 1 - 7.96T + 61T^{2}
67 1+9.61T+67T2 1 + 9.61T + 67T^{2}
71 1−3.23T+71T2 1 - 3.23T + 71T^{2}
73 1+3.87T+73T2 1 + 3.87T + 73T^{2}
79 1−14.8T+79T2 1 - 14.8T + 79T^{2}
83 1−16.2T+83T2 1 - 16.2T + 83T^{2}
89 1+11.9T+89T2 1 + 11.9T + 89T^{2}
97 1−1.99T+97T2 1 - 1.99T + 97T^{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

−9.293648831847178721313222694898, −8.196846225010305128140450673162, −7.73726585307130418169690488358, −6.51368636074897312173208421120, −5.87201125111447772508716987218, −5.21094421176340378482966988309, −4.26181097848993732451652865454, −3.50845740879403620623355165554, −2.57198086039079186503557863220, −1.73408558671891566192624580232, 1.73408558671891566192624580232, 2.57198086039079186503557863220, 3.50845740879403620623355165554, 4.26181097848993732451652865454, 5.21094421176340378482966988309, 5.87201125111447772508716987218, 6.51368636074897312173208421120, 7.73726585307130418169690488358, 8.196846225010305128140450673162, 9.293648831847178721313222694898

Graph of the ZZ-function along the critical line