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Properties

Label 2-1859-1.1-c3-0-375
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  + 3.18·2-s + 9.92·3-s + 2.17·4-s − 1.24·5-s + 31.6·6-s − 27.3·7-s − 18.5·8-s + 71.4·9-s − 3.95·10-s − 11·11-s + 21.5·12-s − 87.1·14-s − 12.3·15-s − 76.6·16-s + 72.7·17-s + 227.·18-s − 104.·19-s − 2.69·20-s − 271.·21-s − 35.0·22-s − 98.5·23-s − 184.·24-s − 123.·25-s + 441.·27-s − 59.3·28-s − 26.0·29-s − 39.2·30-s + ⋯
L(s)  = 1  + 1.12·2-s + 1.90·3-s + 0.271·4-s − 0.110·5-s + 2.15·6-s − 1.47·7-s − 0.821·8-s + 2.64·9-s − 0.125·10-s − 0.301·11-s + 0.519·12-s − 1.66·14-s − 0.211·15-s − 1.19·16-s + 1.03·17-s + 2.98·18-s − 1.25·19-s − 0.0301·20-s − 2.81·21-s − 0.340·22-s − 0.893·23-s − 1.56·24-s − 0.987·25-s + 3.14·27-s − 0.400·28-s − 0.166·29-s − 0.238·30-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 1−3.18T+8T2 1 - 3.18T + 8T^{2}
3 1−9.92T+27T2 1 - 9.92T + 27T^{2}
5 1+1.24T+125T2 1 + 1.24T + 125T^{2}
7 1+27.3T+343T2 1 + 27.3T + 343T^{2}
17 1−72.7T+4.91e3T2 1 - 72.7T + 4.91e3T^{2}
19 1+104.T+6.85e3T2 1 + 104.T + 6.85e3T^{2}
23 1+98.5T+1.21e4T2 1 + 98.5T + 1.21e4T^{2}
29 1+26.0T+2.43e4T2 1 + 26.0T + 2.43e4T^{2}
31 1−170.T+2.97e4T2 1 - 170.T + 2.97e4T^{2}
37 1+340.T+5.06e4T2 1 + 340.T + 5.06e4T^{2}
41 1+261.T+6.89e4T2 1 + 261.T + 6.89e4T^{2}
43 1−10.6T+7.95e4T2 1 - 10.6T + 7.95e4T^{2}
47 1+577.T+1.03e5T2 1 + 577.T + 1.03e5T^{2}
53 1+2.47T+1.48e5T2 1 + 2.47T + 1.48e5T^{2}
59 1−109.T+2.05e5T2 1 - 109.T + 2.05e5T^{2}
61 1+678.T+2.26e5T2 1 + 678.T + 2.26e5T^{2}
67 1+320.T+3.00e5T2 1 + 320.T + 3.00e5T^{2}
71 1−502.T+3.57e5T2 1 - 502.T + 3.57e5T^{2}
73 1−310.T+3.89e5T2 1 - 310.T + 3.89e5T^{2}
79 1−410.T+4.93e5T2 1 - 410.T + 4.93e5T^{2}
83 1−809.T+5.71e5T2 1 - 809.T + 5.71e5T^{2}
89 1−993.T+7.04e5T2 1 - 993.T + 7.04e5T^{2}
97 1+172.T+9.12e5T2 1 + 172.T + 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

−8.436981369866148756181981664121, −7.85768577116099177730752220310, −6.76216587290631287374637543259, −6.19290853101676344433890995480, −4.94632586030325510151073191030, −3.91702601083358788548991065690, −3.49798400302828978822848164586, −2.85195230196389376411278775702, −1.91727274551041571068685054482, 0, 1.91727274551041571068685054482, 2.85195230196389376411278775702, 3.49798400302828978822848164586, 3.91702601083358788548991065690, 4.94632586030325510151073191030, 6.19290853101676344433890995480, 6.76216587290631287374637543259, 7.85768577116099177730752220310, 8.436981369866148756181981664121

Graph of the ZZ-function along the critical line