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Properties

Label 2-363-1.1-c3-0-35
Degree 22
Conductor 363363
Sign −1-1
Analytic cond. 21.417621.4176
Root an. cond. 4.627924.62792
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 7·4-s − 4·5-s − 3·6-s + 26·7-s − 15·8-s + 9·9-s − 4·10-s + 21·12-s + 32·13-s + 26·14-s + 12·15-s + 41·16-s − 74·17-s + 9·18-s + 60·19-s + 28·20-s − 78·21-s − 182·23-s + 45·24-s − 109·25-s + 32·26-s − 27·27-s − 182·28-s + 90·29-s + 12·30-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.357·5-s − 0.204·6-s + 1.40·7-s − 0.662·8-s + 1/3·9-s − 0.126·10-s + 0.505·12-s + 0.682·13-s + 0.496·14-s + 0.206·15-s + 0.640·16-s − 1.05·17-s + 0.117·18-s + 0.724·19-s + 0.313·20-s − 0.810·21-s − 1.64·23-s + 0.382·24-s − 0.871·25-s + 0.241·26-s − 0.192·27-s − 1.22·28-s + 0.576·29-s + 0.0730·30-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 21.417621.4176
Root analytic conductor: 4.627924.62792
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 363, ( :3/2), −1)(2,\ 363,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
11 1 1
good2 1−T+p3T2 1 - T + p^{3} T^{2}
5 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
7 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
13 1−32T+p3T2 1 - 32 T + p^{3} T^{2}
17 1+74T+p3T2 1 + 74 T + p^{3} T^{2}
19 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
23 1+182T+p3T2 1 + 182 T + p^{3} T^{2}
29 1−90T+p3T2 1 - 90 T + p^{3} T^{2}
31 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
37 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
41 1+422T+p3T2 1 + 422 T + p^{3} T^{2}
43 1+408T+p3T2 1 + 408 T + p^{3} T^{2}
47 1+506T+p3T2 1 + 506 T + p^{3} T^{2}
53 1−348T+p3T2 1 - 348 T + p^{3} T^{2}
59 1+200T+p3T2 1 + 200 T + p^{3} T^{2}
61 1+132T+p3T2 1 + 132 T + p^{3} T^{2}
67 1+1036T+p3T2 1 + 1036 T + p^{3} T^{2}
71 1−762T+p3T2 1 - 762 T + p^{3} T^{2}
73 1−542T+p3T2 1 - 542 T + p^{3} T^{2}
79 1−550T+p3T2 1 - 550 T + p^{3} T^{2}
83 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
89 1−570T+p3T2 1 - 570 T + p^{3} T^{2}
97 1−14T+p3T2 1 - 14 T + p^{3} 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

−10.68986559669030505440389264907, −9.685135409559019949928867758367, −8.452522330878210111894997890577, −7.970854895944839855249943744404, −6.49448391590043741937216313361, −5.36671480335449753650669227405, −4.61367754936720743312896877246, −3.70160502417944553440467732930, −1.63346534628870380534544130601, 0, 1.63346534628870380534544130601, 3.70160502417944553440467732930, 4.61367754936720743312896877246, 5.36671480335449753650669227405, 6.49448391590043741937216313361, 7.970854895944839855249943744404, 8.452522330878210111894997890577, 9.685135409559019949928867758367, 10.68986559669030505440389264907

Graph of the ZZ-function along the critical line