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Properties

Label 2-1859-1.1-c3-0-257
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  + 0.641·2-s − 1.91·3-s − 7.58·4-s + 0.642·5-s − 1.22·6-s + 23.2·7-s − 9.99·8-s − 23.3·9-s + 0.411·10-s − 11·11-s + 14.5·12-s + 14.8·14-s − 1.22·15-s + 54.2·16-s − 35.1·17-s − 14.9·18-s + 37.3·19-s − 4.87·20-s − 44.4·21-s − 7.05·22-s + 121.·23-s + 19.1·24-s − 124.·25-s + 96.3·27-s − 176.·28-s − 19.7·29-s − 0.788·30-s + ⋯
L(s)  = 1  + 0.226·2-s − 0.368·3-s − 0.948·4-s + 0.0574·5-s − 0.0834·6-s + 1.25·7-s − 0.441·8-s − 0.864·9-s + 0.0130·10-s − 0.301·11-s + 0.349·12-s + 0.284·14-s − 0.0211·15-s + 0.848·16-s − 0.502·17-s − 0.195·18-s + 0.451·19-s − 0.0545·20-s − 0.461·21-s − 0.0683·22-s + 1.10·23-s + 0.162·24-s − 0.996·25-s + 0.686·27-s − 1.19·28-s − 0.126·29-s − 0.00479·30-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 1−0.641T+8T2 1 - 0.641T + 8T^{2}
3 1+1.91T+27T2 1 + 1.91T + 27T^{2}
5 1−0.642T+125T2 1 - 0.642T + 125T^{2}
7 1−23.2T+343T2 1 - 23.2T + 343T^{2}
17 1+35.1T+4.91e3T2 1 + 35.1T + 4.91e3T^{2}
19 1−37.3T+6.85e3T2 1 - 37.3T + 6.85e3T^{2}
23 1−121.T+1.21e4T2 1 - 121.T + 1.21e4T^{2}
29 1+19.7T+2.43e4T2 1 + 19.7T + 2.43e4T^{2}
31 1−30.1T+2.97e4T2 1 - 30.1T + 2.97e4T^{2}
37 1+193.T+5.06e4T2 1 + 193.T + 5.06e4T^{2}
41 1+80.8T+6.89e4T2 1 + 80.8T + 6.89e4T^{2}
43 1−196.T+7.95e4T2 1 - 196.T + 7.95e4T^{2}
47 1+182.T+1.03e5T2 1 + 182.T + 1.03e5T^{2}
53 1−451.T+1.48e5T2 1 - 451.T + 1.48e5T^{2}
59 1−270.T+2.05e5T2 1 - 270.T + 2.05e5T^{2}
61 1−694.T+2.26e5T2 1 - 694.T + 2.26e5T^{2}
67 1−364.T+3.00e5T2 1 - 364.T + 3.00e5T^{2}
71 1+772.T+3.57e5T2 1 + 772.T + 3.57e5T^{2}
73 1−160.T+3.89e5T2 1 - 160.T + 3.89e5T^{2}
79 1−46.4T+4.93e5T2 1 - 46.4T + 4.93e5T^{2}
83 1+520.T+5.71e5T2 1 + 520.T + 5.71e5T^{2}
89 1−394.T+7.04e5T2 1 - 394.T + 7.04e5T^{2}
97 1+17.2T+9.12e5T2 1 + 17.2T + 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

−8.500076957725515168410184027275, −7.911604923287478156547847355584, −6.88908071039659223806313683153, −5.68291329671982061126703647236, −5.26582578626351439165066833610, −4.55887661666210738648536287701, −3.58288330334626811770985267191, −2.43123965403167903941082102509, −1.12046292717019166173503483044, 0, 1.12046292717019166173503483044, 2.43123965403167903941082102509, 3.58288330334626811770985267191, 4.55887661666210738648536287701, 5.26582578626351439165066833610, 5.68291329671982061126703647236, 6.88908071039659223806313683153, 7.911604923287478156547847355584, 8.500076957725515168410184027275

Graph of the ZZ-function along the critical line