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Properties

Label 2-363-1.1-c1-0-8
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  − 1.73·2-s − 3-s + 0.999·4-s − 3·5-s + 1.73·6-s + 3.46·7-s + 1.73·8-s + 9-s + 5.19·10-s − 0.999·12-s + 1.73·13-s − 5.99·14-s + 3·15-s − 5·16-s − 1.73·17-s − 1.73·18-s − 6.92·19-s − 2.99·20-s − 3.46·21-s − 6·23-s − 1.73·24-s + 4·25-s − 2.99·26-s − 27-s + 3.46·28-s + 1.73·29-s − 5.19·30-s + ⋯
L(s)  = 1  − 1.22·2-s − 0.577·3-s + 0.499·4-s − 1.34·5-s + 0.707·6-s + 1.30·7-s + 0.612·8-s + 0.333·9-s + 1.64·10-s − 0.288·12-s + 0.480·13-s − 1.60·14-s + 0.774·15-s − 1.25·16-s − 0.420·17-s − 0.408·18-s − 1.58·19-s − 0.670·20-s − 0.755·21-s − 1.25·23-s − 0.353·24-s + 0.800·25-s − 0.588·26-s − 0.192·27-s + 0.654·28-s + 0.321·29-s − 0.948·30-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+1.73T+2T2 1 + 1.73T + 2T^{2}
5 1+3T+5T2 1 + 3T + 5T^{2}
7 1−3.46T+7T2 1 - 3.46T + 7T^{2}
13 1−1.73T+13T2 1 - 1.73T + 13T^{2}
17 1+1.73T+17T2 1 + 1.73T + 17T^{2}
19 1+6.92T+19T2 1 + 6.92T + 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1−1.73T+29T2 1 - 1.73T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+11T+37T2 1 + 11T + 37T^{2}
41 1−1.73T+41T2 1 - 1.73T + 41T^{2}
43 1+3.46T+43T2 1 + 3.46T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+9T+53T2 1 + 9T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1+2T+67T2 1 + 2T + 67T^{2}
71 1+6T+71T2 1 + 6T + 71T^{2}
73 1+6.92T+73T2 1 + 6.92T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−9T+89T2 1 - 9T + 89T^{2}
97 1+7T+97T2 1 + 7T + 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

−10.88929931871862449135005452623, −10.28328728028678558856477572353, −8.789592707949681467230495188896, −8.233106999068815384359466088835, −7.59634104912019193486672986525, −6.45942841117226412127905637665, −4.78736398400057893106922846764, −4.08874146042552895871178962658, −1.70975558990753806142579652840, 0, 1.70975558990753806142579652840, 4.08874146042552895871178962658, 4.78736398400057893106922846764, 6.45942841117226412127905637665, 7.59634104912019193486672986525, 8.233106999068815384359466088835, 8.789592707949681467230495188896, 10.28328728028678558856477572353, 10.88929931871862449135005452623

Graph of the ZZ-function along the critical line