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Properties

Label 2-1859-1.1-c3-0-162
Degree 22
Conductor 18591859
Sign −1-1
Analytic cond. 109.684109.684
Root an. cond. 10.473010.4730
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.532·2-s + 1.85·3-s − 7.71·4-s − 18.1·5-s − 0.988·6-s − 20.2·7-s + 8.37·8-s − 23.5·9-s + 9.64·10-s − 11·11-s − 14.3·12-s + 10.8·14-s − 33.6·15-s + 57.2·16-s − 66.4·17-s + 12.5·18-s + 139.·19-s + 139.·20-s − 37.6·21-s + 5.86·22-s − 30.4·23-s + 15.5·24-s + 202.·25-s − 93.8·27-s + 156.·28-s + 195.·29-s + 17.9·30-s + ⋯
L(s)  = 1  − 0.188·2-s + 0.357·3-s − 0.964·4-s − 1.61·5-s − 0.0672·6-s − 1.09·7-s + 0.370·8-s − 0.872·9-s + 0.305·10-s − 0.301·11-s − 0.344·12-s + 0.206·14-s − 0.578·15-s + 0.894·16-s − 0.948·17-s + 0.164·18-s + 1.68·19-s + 1.56·20-s − 0.391·21-s + 0.0567·22-s − 0.275·23-s + 0.132·24-s + 1.62·25-s − 0.668·27-s + 1.05·28-s + 1.25·29-s + 0.108·30-s + ⋯

Functional equation

Λ(s)=(1859s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1859s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18591859    =    11⋅13211 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 109.684109.684
Root analytic conductor: 10.473010.4730
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1859, ( :3/2), −1)(2,\ 1859,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad11 1+11T 1 + 11T
13 1 1
good2 1+0.532T+8T2 1 + 0.532T + 8T^{2}
3 1−1.85T+27T2 1 - 1.85T + 27T^{2}
5 1+18.1T+125T2 1 + 18.1T + 125T^{2}
7 1+20.2T+343T2 1 + 20.2T + 343T^{2}
17 1+66.4T+4.91e3T2 1 + 66.4T + 4.91e3T^{2}
19 1−139.T+6.85e3T2 1 - 139.T + 6.85e3T^{2}
23 1+30.4T+1.21e4T2 1 + 30.4T + 1.21e4T^{2}
29 1−195.T+2.43e4T2 1 - 195.T + 2.43e4T^{2}
31 1+70.0T+2.97e4T2 1 + 70.0T + 2.97e4T^{2}
37 1−216.T+5.06e4T2 1 - 216.T + 5.06e4T^{2}
41 1−450.T+6.89e4T2 1 - 450.T + 6.89e4T^{2}
43 1+326.T+7.95e4T2 1 + 326.T + 7.95e4T^{2}
47 1−306.T+1.03e5T2 1 - 306.T + 1.03e5T^{2}
53 1+582.T+1.48e5T2 1 + 582.T + 1.48e5T^{2}
59 1+351.T+2.05e5T2 1 + 351.T + 2.05e5T^{2}
61 1+129.T+2.26e5T2 1 + 129.T + 2.26e5T^{2}
67 1−153.T+3.00e5T2 1 - 153.T + 3.00e5T^{2}
71 1−1.06e3T+3.57e5T2 1 - 1.06e3T + 3.57e5T^{2}
73 1−138.T+3.89e5T2 1 - 138.T + 3.89e5T^{2}
79 1−1.12e3T+4.93e5T2 1 - 1.12e3T + 4.93e5T^{2}
83 1+377.T+5.71e5T2 1 + 377.T + 5.71e5T^{2}
89 1+1.33e3T+7.04e5T2 1 + 1.33e3T + 7.04e5T^{2}
97 1+1.42e3T+9.12e5T2 1 + 1.42e3T + 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

−8.355487759430074941867476806324, −7.956266030733244469471351806083, −7.16350284435565007182830288521, −6.10788491227043917330158650222, −5.05249488509563873209124564966, −4.20542529815843598649384143772, −3.41700306876626067316898196691, −2.82041454344266306958781233518, −0.75831004685198670497979546132, 0, 0.75831004685198670497979546132, 2.82041454344266306958781233518, 3.41700306876626067316898196691, 4.20542529815843598649384143772, 5.05249488509563873209124564966, 6.10788491227043917330158650222, 7.16350284435565007182830288521, 7.956266030733244469471351806083, 8.355487759430074941867476806324

Graph of the ZZ-function along the critical line