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Properties

Label 2-363-1.1-c5-0-54
Degree 22
Conductor 363363
Sign −1-1
Analytic cond. 58.219358.2193
Root an. cond. 7.630157.63015
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s − 9·3-s + 49·4-s + 24·5-s + 81·6-s + 72·7-s − 153·8-s + 81·9-s − 216·10-s − 441·12-s + 306·13-s − 648·14-s − 216·15-s − 191·16-s + 1.20e3·17-s − 729·18-s − 774·19-s + 1.17e3·20-s − 648·21-s − 4.62e3·23-s + 1.37e3·24-s − 2.54e3·25-s − 2.75e3·26-s − 729·27-s + 3.52e3·28-s − 7.68e3·29-s + 1.94e3·30-s + ⋯
L(s)  = 1  − 1.59·2-s − 0.577·3-s + 1.53·4-s + 0.429·5-s + 0.918·6-s + 0.555·7-s − 0.845·8-s + 1/3·9-s − 0.683·10-s − 0.884·12-s + 0.502·13-s − 0.883·14-s − 0.247·15-s − 0.186·16-s + 1.01·17-s − 0.530·18-s − 0.491·19-s + 0.657·20-s − 0.320·21-s − 1.82·23-s + 0.487·24-s − 0.815·25-s − 0.798·26-s − 0.192·27-s + 0.850·28-s − 1.69·29-s + 0.394·30-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 58.219358.2193
Root analytic conductor: 7.630157.63015
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 363, ( :5/2), −1)(2,\ 363,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+p2T 1 + p^{2} T
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

−9.946137289658365258593372063095, −9.457687897137528402867318932394, −8.157860603792140774304566615747, −7.72863711334436395479713052566, −6.42345153254255051011800912543, −5.60696348439850882561103453371, −4.08430797528903131527281778452, −2.15208152315674811919420244456, −1.24259946185683390241928148054, 0, 1.24259946185683390241928148054, 2.15208152315674811919420244456, 4.08430797528903131527281778452, 5.60696348439850882561103453371, 6.42345153254255051011800912543, 7.72863711334436395479713052566, 8.157860603792140774304566615747, 9.457687897137528402867318932394, 9.946137289658365258593372063095

Graph of the ZZ-function along the critical line