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Properties

Label 2-1859-1.1-c3-0-56
Degree 22
Conductor 18591859
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·2-s − 9.62·3-s − 3.22·4-s + 10.6·5-s − 21.0·6-s − 26.7·7-s − 24.5·8-s + 65.6·9-s + 23.3·10-s + 11·11-s + 31.0·12-s − 58.3·14-s − 102.·15-s − 27.7·16-s + 123.·17-s + 143.·18-s + 6.83·19-s − 34.4·20-s + 257.·21-s + 24.0·22-s − 164.·23-s + 236.·24-s − 10.5·25-s − 372.·27-s + 86.1·28-s − 184.·29-s − 224.·30-s + ⋯
L(s)  = 1  + 0.772·2-s − 1.85·3-s − 0.403·4-s + 0.956·5-s − 1.43·6-s − 1.44·7-s − 1.08·8-s + 2.43·9-s + 0.739·10-s + 0.301·11-s + 0.746·12-s − 1.11·14-s − 1.77·15-s − 0.434·16-s + 1.76·17-s + 1.87·18-s + 0.0825·19-s − 0.385·20-s + 2.67·21-s + 0.232·22-s − 1.49·23-s + 2.00·24-s − 0.0847·25-s − 2.65·27-s + 0.581·28-s − 1.17·29-s − 1.36·30-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: 11
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: 00
Selberg data: (2, 1859, ( :3/2), 1)(2,\ 1859,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.66918437420.6691843742
L(12)L(\frac12) ≈\approx 0.66918437420.6691843742
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 1−2.18T+8T2 1 - 2.18T + 8T^{2}
3 1+9.62T+27T2 1 + 9.62T + 27T^{2}
5 1−10.6T+125T2 1 - 10.6T + 125T^{2}
7 1+26.7T+343T2 1 + 26.7T + 343T^{2}
17 1−123.T+4.91e3T2 1 - 123.T + 4.91e3T^{2}
19 1−6.83T+6.85e3T2 1 - 6.83T + 6.85e3T^{2}
23 1+164.T+1.21e4T2 1 + 164.T + 1.21e4T^{2}
29 1+184.T+2.43e4T2 1 + 184.T + 2.43e4T^{2}
31 1+90.4T+2.97e4T2 1 + 90.4T + 2.97e4T^{2}
37 1−98.3T+5.06e4T2 1 - 98.3T + 5.06e4T^{2}
41 1+313.T+6.89e4T2 1 + 313.T + 6.89e4T^{2}
43 1+60.0T+7.95e4T2 1 + 60.0T + 7.95e4T^{2}
47 1+601.T+1.03e5T2 1 + 601.T + 1.03e5T^{2}
53 1+106.T+1.48e5T2 1 + 106.T + 1.48e5T^{2}
59 1+128.T+2.05e5T2 1 + 128.T + 2.05e5T^{2}
61 1+102.T+2.26e5T2 1 + 102.T + 2.26e5T^{2}
67 1+186.T+3.00e5T2 1 + 186.T + 3.00e5T^{2}
71 1−736.T+3.57e5T2 1 - 736.T + 3.57e5T^{2}
73 1−814.T+3.89e5T2 1 - 814.T + 3.89e5T^{2}
79 1−171.T+4.93e5T2 1 - 171.T + 4.93e5T^{2}
83 1+15.5T+5.71e5T2 1 + 15.5T + 5.71e5T^{2}
89 1+878.T+7.04e5T2 1 + 878.T + 7.04e5T^{2}
97 1−750.T+9.12e5T2 1 - 750.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

−9.456634985870976668680172959781, −7.87922151290882562797969250153, −6.66254397456577722476569833373, −6.22872227699418230299831382750, −5.59935812021300691151328070380, −5.21957420480317638160172957712, −4.01246405770842970339472136162, −3.33156614491702784762856759879, −1.65806484141682363241231841276, −0.37758427889913755387502725725, 0.37758427889913755387502725725, 1.65806484141682363241231841276, 3.33156614491702784762856759879, 4.01246405770842970339472136162, 5.21957420480317638160172957712, 5.59935812021300691151328070380, 6.22872227699418230299831382750, 6.66254397456577722476569833373, 7.87922151290882562797969250153, 9.456634985870976668680172959781

Graph of the ZZ-function along the critical line