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Properties

Label 2-363-1.1-c1-0-7
Degree 22
Conductor 363363
Sign −1-1
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s − 3-s + 4.85·4-s − 0.618·5-s + 2.61·6-s − 7-s − 7.47·8-s + 9-s + 1.61·10-s − 4.85·12-s + 0.236·13-s + 2.61·14-s + 0.618·15-s + 9.85·16-s − 1.14·17-s − 2.61·18-s + 5.85·19-s − 3.00·20-s + 21-s + 0.236·23-s + 7.47·24-s − 4.61·25-s − 0.618·26-s − 27-s − 4.85·28-s − 6·29-s − 1.61·30-s + ⋯
L(s)  = 1  − 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.276·5-s + 1.06·6-s − 0.377·7-s − 2.64·8-s + 0.333·9-s + 0.511·10-s − 1.40·12-s + 0.0654·13-s + 0.699·14-s + 0.159·15-s + 2.46·16-s − 0.277·17-s − 0.617·18-s + 1.34·19-s − 0.670·20-s + 0.218·21-s + 0.0492·23-s + 1.52·24-s − 0.923·25-s − 0.121·26-s − 0.192·27-s − 0.917·28-s − 1.11·29-s − 0.295·30-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 363, ( :1/2), −1)(2,\ 363,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
11 1 1
good2 1+2.61T+2T2 1 + 2.61T + 2T^{2}
5 1+0.618T+5T2 1 + 0.618T + 5T^{2}
7 1+T+7T2 1 + T + 7T^{2}
13 1−0.236T+13T2 1 - 0.236T + 13T^{2}
17 1+1.14T+17T2 1 + 1.14T + 17T^{2}
19 1−5.85T+19T2 1 - 5.85T + 19T^{2}
23 1−0.236T+23T2 1 - 0.236T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+6.09T+31T2 1 + 6.09T + 31T^{2}
37 1+6.23T+37T2 1 + 6.23T + 37T^{2}
41 1−0.236T+41T2 1 - 0.236T + 41T^{2}
43 1+6.70T+43T2 1 + 6.70T + 43T^{2}
47 1+10.0T+47T2 1 + 10.0T + 47T^{2}
53 1+0.381T+53T2 1 + 0.381T + 53T^{2}
59 1−7.38T+59T2 1 - 7.38T + 59T^{2}
61 1+11.5T+61T2 1 + 11.5T + 61T^{2}
67 1−1.85T+67T2 1 - 1.85T + 67T^{2}
71 1−10.3T+71T2 1 - 10.3T + 71T^{2}
73 1+5.70T+73T2 1 + 5.70T + 73T^{2}
79 1−11T+79T2 1 - 11T + 79T^{2}
83 1−1.47T+83T2 1 - 1.47T + 83T^{2}
89 1+8.23T+89T2 1 + 8.23T + 89T^{2}
97 1−7.85T+97T2 1 - 7.85T + 97T^{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

−10.87953420496836069222112139210, −9.827318863042998182734568494876, −9.347426769669901067521771697937, −8.192667823969700711798817326454, −7.36964410241537205204447272507, −6.57840427014371539352422085539, −5.42086985009428714237368537543, −3.41652021169201269082197428946, −1.69987514028985068024643792656, 0, 1.69987514028985068024643792656, 3.41652021169201269082197428946, 5.42086985009428714237368537543, 6.57840427014371539352422085539, 7.36964410241537205204447272507, 8.192667823969700711798817326454, 9.347426769669901067521771697937, 9.827318863042998182734568494876, 10.87953420496836069222112139210

Graph of the ZZ-function along the critical line