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Properties

Label 2-363-1.1-c1-0-1
Degree 22
Conductor 363363
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.52·2-s + 3-s + 4.37·4-s − 2.37·5-s − 2.52·6-s − 0.792·7-s − 5.98·8-s + 9-s + 5.98·10-s + 4.37·12-s + 4.10·13-s + 2·14-s − 2.37·15-s + 6.37·16-s + 5.98·17-s − 2.52·18-s − 4.25·19-s − 10.3·20-s − 0.792·21-s + 2·23-s − 5.98·24-s + 0.627·25-s − 10.3·26-s + 27-s − 3.46·28-s + 2.52·29-s + 5.98·30-s + ⋯
L(s)  = 1  − 1.78·2-s + 0.577·3-s + 2.18·4-s − 1.06·5-s − 1.03·6-s − 0.299·7-s − 2.11·8-s + 0.333·9-s + 1.89·10-s + 1.26·12-s + 1.13·13-s + 0.534·14-s − 0.612·15-s + 1.59·16-s + 1.45·17-s − 0.594·18-s − 0.976·19-s − 2.31·20-s − 0.172·21-s + 0.417·23-s − 1.22·24-s + 0.125·25-s − 2.03·26-s + 0.192·27-s − 0.654·28-s + 0.468·29-s + 1.09·30-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: 11
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: 00
Selberg data: (2, 363, ( :1/2), 1)(2,\ 363,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.61923160750.6192316075
L(12)L(\frac12) ≈\approx 0.61923160750.6192316075
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+2.52T+2T2 1 + 2.52T + 2T^{2}
5 1+2.37T+5T2 1 + 2.37T + 5T^{2}
7 1+0.792T+7T2 1 + 0.792T + 7T^{2}
13 1−4.10T+13T2 1 - 4.10T + 13T^{2}
17 1−5.98T+17T2 1 - 5.98T + 17T^{2}
19 1+4.25T+19T2 1 + 4.25T + 19T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1−2.52T+29T2 1 - 2.52T + 29T^{2}
31 1−7.37T+31T2 1 - 7.37T + 31T^{2}
37 1−5T+37T2 1 - 5T + 37T^{2}
41 1−5.69T+41T2 1 - 5.69T + 41T^{2}
43 1−6.63T+43T2 1 - 6.63T + 43T^{2}
47 1+1.25T+47T2 1 + 1.25T + 47T^{2}
53 1−13.1T+53T2 1 - 13.1T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1−2.67T+61T2 1 - 2.67T + 61T^{2}
67 1−16.1T+67T2 1 - 16.1T + 67T^{2}
71 1−0.744T+71T2 1 - 0.744T + 71T^{2}
73 1+7.42T+73T2 1 + 7.42T + 73T^{2}
79 1+5.84T+79T2 1 + 5.84T + 79T^{2}
83 1+8.51T+83T2 1 + 8.51T + 83T^{2}
89 1+6.37T+89T2 1 + 6.37T + 89T^{2}
97 1+12.4T+97T2 1 + 12.4T + 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

−11.14482558751290845869326472592, −10.30861232147378039791530806874, −9.478668154943856534217489353053, −8.447189326746436467389406982449, −8.082886010611771925236573919365, −7.17610448933405091698685460594, −6.13183667544622170210800127864, −4.01586908165467332267389728373, −2.78563178180183564246620900160, −1.01647366498656575723404249336, 1.01647366498656575723404249336, 2.78563178180183564246620900160, 4.01586908165467332267389728373, 6.13183667544622170210800127864, 7.17610448933405091698685460594, 8.082886010611771925236573919365, 8.447189326746436467389406982449, 9.478668154943856534217489353053, 10.30861232147378039791530806874, 11.14482558751290845869326472592

Graph of the ZZ-function along the critical line