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Properties

Label 2-1859-1.1-c3-0-225
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  − 4.83·2-s − 7.08·3-s + 15.3·4-s + 9.07·5-s + 34.2·6-s + 2.15·7-s − 35.4·8-s + 23.2·9-s − 43.8·10-s − 11·11-s − 108.·12-s − 10.4·14-s − 64.3·15-s + 48.4·16-s + 49.8·17-s − 112.·18-s + 95.3·19-s + 139.·20-s − 15.2·21-s + 53.1·22-s − 96.0·23-s + 251.·24-s − 42.6·25-s + 26.6·27-s + 33.0·28-s + 115.·29-s + 310.·30-s + ⋯
L(s)  = 1  − 1.70·2-s − 1.36·3-s + 1.91·4-s + 0.811·5-s + 2.32·6-s + 0.116·7-s − 1.56·8-s + 0.860·9-s − 1.38·10-s − 0.301·11-s − 2.61·12-s − 0.198·14-s − 1.10·15-s + 0.756·16-s + 0.711·17-s − 1.47·18-s + 1.15·19-s + 1.55·20-s − 0.158·21-s + 0.514·22-s − 0.871·23-s + 2.13·24-s − 0.341·25-s + 0.189·27-s + 0.223·28-s + 0.736·29-s + 1.89·30-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+4.83T+8T2 1 + 4.83T + 8T^{2}
3 1+7.08T+27T2 1 + 7.08T + 27T^{2}
5 1−9.07T+125T2 1 - 9.07T + 125T^{2}
7 1−2.15T+343T2 1 - 2.15T + 343T^{2}
17 1−49.8T+4.91e3T2 1 - 49.8T + 4.91e3T^{2}
19 1−95.3T+6.85e3T2 1 - 95.3T + 6.85e3T^{2}
23 1+96.0T+1.21e4T2 1 + 96.0T + 1.21e4T^{2}
29 1−115.T+2.43e4T2 1 - 115.T + 2.43e4T^{2}
31 1−78.6T+2.97e4T2 1 - 78.6T + 2.97e4T^{2}
37 1+13.4T+5.06e4T2 1 + 13.4T + 5.06e4T^{2}
41 1+122.T+6.89e4T2 1 + 122.T + 6.89e4T^{2}
43 1−95.5T+7.95e4T2 1 - 95.5T + 7.95e4T^{2}
47 1+361.T+1.03e5T2 1 + 361.T + 1.03e5T^{2}
53 1−617.T+1.48e5T2 1 - 617.T + 1.48e5T^{2}
59 1+723.T+2.05e5T2 1 + 723.T + 2.05e5T^{2}
61 1+549.T+2.26e5T2 1 + 549.T + 2.26e5T^{2}
67 1−554.T+3.00e5T2 1 - 554.T + 3.00e5T^{2}
71 1−99.2T+3.57e5T2 1 - 99.2T + 3.57e5T^{2}
73 1+609.T+3.89e5T2 1 + 609.T + 3.89e5T^{2}
79 1+586.T+4.93e5T2 1 + 586.T + 4.93e5T^{2}
83 1+5.23T+5.71e5T2 1 + 5.23T + 5.71e5T^{2}
89 1+789.T+7.04e5T2 1 + 789.T + 7.04e5T^{2}
97 1+1.65e3T+9.12e5T2 1 + 1.65e3T + 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

−8.515263003155460836581071162364, −7.78397921241041712740274353439, −6.99510760539184346282375633562, −6.16014872786719887786352392164, −5.64480975434470692685012444863, −4.69803408327296558717899571023, −3.00034100339696436288970918688, −1.78289929543610708464300869293, −0.988319188236137167928241106737, 0, 0.988319188236137167928241106737, 1.78289929543610708464300869293, 3.00034100339696436288970918688, 4.69803408327296558717899571023, 5.64480975434470692685012444863, 6.16014872786719887786352392164, 6.99510760539184346282375633562, 7.78397921241041712740274353439, 8.515263003155460836581071162364

Graph of the ZZ-function along the critical line