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Properties

Label 2-363-33.20-c0-0-0
Degree 22
Conductor 363363
Sign 0.569+0.821i0.569 + 0.821i
Analytic cond. 0.1811600.181160
Root an. cond. 0.4256290.425629
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)9-s − 12-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.618 + 1.90i)31-s + (−0.809 + 0.587i)36-s + (1.61 + 1.17i)37-s + (0.809 + 0.587i)48-s + (−0.309 − 0.951i)49-s + (0.309 − 0.951i)64-s − 2·67-s + (−0.309 + 0.951i)75-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)9-s − 12-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.618 + 1.90i)31-s + (−0.809 + 0.587i)36-s + (1.61 + 1.17i)37-s + (0.809 + 0.587i)48-s + (−0.309 − 0.951i)49-s + (0.309 − 0.951i)64-s − 2·67-s + (−0.309 + 0.951i)75-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.569+0.821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s) L(s)=((0.569+0.821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.569+0.821i0.569 + 0.821i
Analytic conductor: 0.1811600.181160
Root analytic conductor: 0.4256290.425629
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ363(251,⋅)\chi_{363} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :0), 0.569+0.821i)(2,\ 363,\ (\ :0),\ 0.569 + 0.821i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87873959090.8787395909
L(12)L(\frac12) ≈\approx 0.87873959090.8787395909
L(1)L(1) 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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
11 1 1
good2 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
5 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
7 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
13 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
17 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
19 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
23 1−T2 1 - T^{2}
29 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
31 1+(0.618−1.90i)T+(−0.809−0.587i)T2 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2}
37 1+(−1.61−1.17i)T+(0.309+0.951i)T2 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2}
41 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
43 1+T2 1 + T^{2}
47 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
67 1+2T+T2 1 + 2T + T^{2}
71 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
73 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
79 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1−T2 1 - T^{2}
97 1+(−0.618+1.90i)T+(−0.809−0.587i)T2 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)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

−11.60334923423243049189003682351, −10.35947963128729202997002505678, −9.525875442102522616499636580942, −8.769073372381244744378607124484, −7.925887670773080368600918124268, −6.80864664172881355213017516475, −5.71398631229508462563979728516, −4.45532730895249826227240071082, −3.23112915873196011125024679814, −1.54848778935769911800562078805, 2.51911550497708430071669515855, 3.81368417415172572109019823283, 4.51339613770582073786590703460, 5.81687527139242650830957661593, 7.55258136841995786417402856092, 8.068266425726644965230629538474, 9.189110537970495911476189432514, 9.612937094737320800479932744133, 10.73428730324520278723532565357, 11.82127054859491686278185255043

Graph of the ZZ-function along the critical line