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Properties

Label 2-1859-1.1-c3-0-300
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  − 2.83·2-s + 8.17·3-s + 0.0460·4-s − 2.30·5-s − 23.1·6-s + 3.15·7-s + 22.5·8-s + 39.8·9-s + 6.55·10-s − 11·11-s + 0.376·12-s − 8.95·14-s − 18.8·15-s − 64.3·16-s − 127.·17-s − 113.·18-s + 108.·19-s − 0.106·20-s + 25.8·21-s + 31.2·22-s − 19.0·23-s + 184.·24-s − 119.·25-s + 105.·27-s + 0.145·28-s + 197.·29-s + 53.5·30-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.57·3-s + 0.00576·4-s − 0.206·5-s − 1.57·6-s + 0.170·7-s + 0.997·8-s + 1.47·9-s + 0.207·10-s − 0.301·11-s + 0.00906·12-s − 0.171·14-s − 0.325·15-s − 1.00·16-s − 1.82·17-s − 1.48·18-s + 1.30·19-s − 0.00118·20-s + 0.268·21-s + 0.302·22-s − 0.172·23-s + 1.56·24-s − 0.957·25-s + 0.749·27-s + 0.000982·28-s + 1.26·29-s + 0.325·30-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+2.83T+8T2 1 + 2.83T + 8T^{2}
3 1−8.17T+27T2 1 - 8.17T + 27T^{2}
5 1+2.30T+125T2 1 + 2.30T + 125T^{2}
7 1−3.15T+343T2 1 - 3.15T + 343T^{2}
17 1+127.T+4.91e3T2 1 + 127.T + 4.91e3T^{2}
19 1−108.T+6.85e3T2 1 - 108.T + 6.85e3T^{2}
23 1+19.0T+1.21e4T2 1 + 19.0T + 1.21e4T^{2}
29 1−197.T+2.43e4T2 1 - 197.T + 2.43e4T^{2}
31 1−262.T+2.97e4T2 1 - 262.T + 2.97e4T^{2}
37 1+213.T+5.06e4T2 1 + 213.T + 5.06e4T^{2}
41 1+227.T+6.89e4T2 1 + 227.T + 6.89e4T^{2}
43 1+213.T+7.95e4T2 1 + 213.T + 7.95e4T^{2}
47 1−326.T+1.03e5T2 1 - 326.T + 1.03e5T^{2}
53 1+137.T+1.48e5T2 1 + 137.T + 1.48e5T^{2}
59 1−234.T+2.05e5T2 1 - 234.T + 2.05e5T^{2}
61 1+406.T+2.26e5T2 1 + 406.T + 2.26e5T^{2}
67 1+625.T+3.00e5T2 1 + 625.T + 3.00e5T^{2}
71 1+399.T+3.57e5T2 1 + 399.T + 3.57e5T^{2}
73 1+583.T+3.89e5T2 1 + 583.T + 3.89e5T^{2}
79 1−578.T+4.93e5T2 1 - 578.T + 4.93e5T^{2}
83 1+103.T+5.71e5T2 1 + 103.T + 5.71e5T^{2}
89 1−660.T+7.04e5T2 1 - 660.T + 7.04e5T^{2}
97 1+1.18e3T+9.12e5T2 1 + 1.18e3T + 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

−8.647560231611616672387677233888, −7.946420339571388207867923199666, −7.40045109058576079872149631763, −6.50006366213834006974200185846, −4.92598468981180686210154266998, −4.24873587661639734397123478588, −3.20444003096770659410597714698, −2.27986668673898367855715885948, −1.36381245306145200242025705524, 0, 1.36381245306145200242025705524, 2.27986668673898367855715885948, 3.20444003096770659410597714698, 4.24873587661639734397123478588, 4.92598468981180686210154266998, 6.50006366213834006974200185846, 7.40045109058576079872149631763, 7.946420339571388207867923199666, 8.647560231611616672387677233888

Graph of the ZZ-function along the critical line