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Properties

Label 2-363-1.1-c1-0-3
Degree 22
Conductor 363363
Sign 11
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 4·5-s + 2·6-s + 7-s + 9-s − 8·10-s − 2·12-s − 2·13-s − 2·14-s − 4·15-s − 4·16-s + 4·17-s − 2·18-s − 3·19-s + 8·20-s − 21-s + 2·23-s + 11·25-s + 4·26-s − 27-s + 2·28-s + 6·29-s + 8·30-s − 5·31-s + 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 1.78·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 2.52·10-s − 0.577·12-s − 0.554·13-s − 0.534·14-s − 1.03·15-s − 16-s + 0.970·17-s − 0.471·18-s − 0.688·19-s + 1.78·20-s − 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.46·30-s − 0.898·31-s + 1.41·32-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: 11
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), 1)(2,\ 363,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.78617357600.7861735760
L(12)L(\frac12) ≈\approx 0.78617357600.7861735760
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+pT+pT2 1 + p T + p T^{2}
5 1−4T+pT2 1 - 4 T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+3T+pT2 1 + 3 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−3T+pT2 1 - 3 T + p T^{2}
67 1+T+pT2 1 + T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−5T+pT2 1 - 5 T + p 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

−10.86340229289232847766758241404, −10.38252022530035955702643197808, −9.598150501746471856910168767253, −8.972019134073409502580254099142, −7.79298545455764963624452711488, −6.75395532990341575165448572479, −5.80045457911231193627474837008, −4.79085093103530693850477639954, −2.38116356225299620011633985249, −1.22560267668749840643223616905, 1.22560267668749840643223616905, 2.38116356225299620011633985249, 4.79085093103530693850477639954, 5.80045457911231193627474837008, 6.75395532990341575165448572479, 7.79298545455764963624452711488, 8.972019134073409502580254099142, 9.598150501746471856910168767253, 10.38252022530035955702643197808, 10.86340229289232847766758241404

Graph of the ZZ-function along the critical line