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Properties

Label 2-33e2-1.1-c5-0-39
Degree 22
Conductor 10891089
Sign 11
Analytic cond. 174.657174.657
Root an. cond. 13.215813.2158
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s + 49·4-s − 24·5-s − 72·7-s − 153·8-s + 216·10-s − 306·13-s + 648·14-s − 191·16-s + 1.20e3·17-s + 774·19-s − 1.17e3·20-s + 4.62e3·23-s − 2.54e3·25-s + 2.75e3·26-s − 3.52e3·28-s − 7.68e3·29-s + 5.42e3·31-s + 6.61e3·32-s − 1.08e4·34-s + 1.72e3·35-s + 3.45e3·37-s − 6.96e3·38-s + 3.67e3·40-s + 7.86e3·41-s − 1.57e4·43-s − 4.16e4·46-s + ⋯
L(s)  = 1  − 1.59·2-s + 1.53·4-s − 0.429·5-s − 0.555·7-s − 0.845·8-s + 0.683·10-s − 0.502·13-s + 0.883·14-s − 0.186·16-s + 1.01·17-s + 0.491·19-s − 0.657·20-s + 1.82·23-s − 0.815·25-s + 0.798·26-s − 0.850·28-s − 1.69·29-s + 1.01·31-s + 1.14·32-s − 1.61·34-s + 0.238·35-s + 0.414·37-s − 0.782·38-s + 0.362·40-s + 0.730·41-s − 1.30·43-s − 2.90·46-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 174.657174.657
Root analytic conductor: 13.215813.2158
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1089, ( :5/2), 1)(2,\ 1089,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.62730137980.6273013798
L(12)L(\frac12) ≈\approx 0.62730137980.6273013798
L(72)L(\frac{7}{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)
bad3 1 1
11 1 1
good2 1+9T+p5T2 1 + 9 T + p^{5} T^{2}
5 1+24T+p5T2 1 + 24 T + p^{5} T^{2}
7 1+72T+p5T2 1 + 72 T + p^{5} T^{2}
13 1+306T+p5T2 1 + 306 T + p^{5} T^{2}
17 1−1206T+p5T2 1 - 1206 T + p^{5} T^{2}
19 1−774T+p5T2 1 - 774 T + p^{5} T^{2}
23 1−4626T+p5T2 1 - 4626 T + p^{5} T^{2}
29 1+7686T+p5T2 1 + 7686 T + p^{5} T^{2}
31 1−5428T+p5T2 1 - 5428 T + p^{5} T^{2}
37 1−3454T+p5T2 1 - 3454 T + p^{5} T^{2}
41 1−7866T+p5T2 1 - 7866 T + p^{5} T^{2}
43 1+15786T+p5T2 1 + 15786 T + p^{5} T^{2}
47 1−6402T+p5T2 1 - 6402 T + p^{5} T^{2}
53 1−21684T+p5T2 1 - 21684 T + p^{5} T^{2}
59 1−27420T+p5T2 1 - 27420 T + p^{5} T^{2}
61 1+52866T+p5T2 1 + 52866 T + p^{5} T^{2}
67 1−25012T+p5T2 1 - 25012 T + p^{5} T^{2}
71 1+65058T+p5T2 1 + 65058 T + p^{5} T^{2}
73 1−26676T+p5T2 1 - 26676 T + p^{5} T^{2}
79 1−18612T+p5T2 1 - 18612 T + p^{5} T^{2}
83 1+p5T2 1 + p^{5} T^{2}
89 1−41670T+p5T2 1 - 41670 T + p^{5} T^{2}
97 1−40694T+p5T2 1 - 40694 T + p^{5} T^{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

−9.289833229140719624289387716241, −8.372487584723894477886109250807, −7.53886410345486398164154002407, −7.14240269828144831204748173683, −6.04578073254428847950718010378, −4.92751451389878745695036803798, −3.58339286367221452245573353103, −2.61704080710370944423389283394, −1.36936672929772883105169912397, −0.46906297954454757541592899878, 0.46906297954454757541592899878, 1.36936672929772883105169912397, 2.61704080710370944423389283394, 3.58339286367221452245573353103, 4.92751451389878745695036803798, 6.04578073254428847950718010378, 7.14240269828144831204748173683, 7.53886410345486398164154002407, 8.372487584723894477886109250807, 9.289833229140719624289387716241

Graph of the ZZ-function along the critical line