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Properties

Label 2-117-3.2-c2-0-5
Degree 22
Conductor 117117
Sign 0.577+0.816i0.577 + 0.816i
Analytic cond. 3.188013.18801
Root an. cond. 1.785501.78550
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.238i·2-s + 3.94·4-s − 9.50i·5-s − 5.89·7-s + 1.89i·8-s + 2.26·10-s − 9.02i·11-s + 3.60·13-s − 1.40i·14-s + 15.3·16-s + 19.8i·17-s + 32.2·19-s − 37.4i·20-s + 2.15·22-s + 2.25i·23-s + ⋯
L(s)  = 1  + 0.119i·2-s + 0.985·4-s − 1.90i·5-s − 0.841·7-s + 0.236i·8-s + 0.226·10-s − 0.820i·11-s + 0.277·13-s − 0.100i·14-s + 0.957·16-s + 1.16i·17-s + 1.69·19-s − 1.87i·20-s + 0.0979·22-s + 0.0979i·23-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.577+0.816i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.577+0.816i0.577 + 0.816i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(53,⋅)\chi_{117} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), 0.577+0.816i)(2,\ 117,\ (\ :1),\ 0.577 + 0.816i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.38544−0.717160i1.38544 - 0.717160i
L(12)L(\frac12) ≈\approx 1.38544−0.717160i1.38544 - 0.717160i
L(2)L(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 1
13 1−3.60T 1 - 3.60T
good2 1−0.238iT−4T2 1 - 0.238iT - 4T^{2}
5 1+9.50iT−25T2 1 + 9.50iT - 25T^{2}
7 1+5.89T+49T2 1 + 5.89T + 49T^{2}
11 1+9.02iT−121T2 1 + 9.02iT - 121T^{2}
17 1−19.8iT−289T2 1 - 19.8iT - 289T^{2}
19 1−32.2T+361T2 1 - 32.2T + 361T^{2}
23 1−2.25iT−529T2 1 - 2.25iT - 529T^{2}
29 1−25.2iT−841T2 1 - 25.2iT - 841T^{2}
31 1−8.89T+961T2 1 - 8.89T + 961T^{2}
37 1+38.8T+1.36e3T2 1 + 38.8T + 1.36e3T^{2}
41 1−29.5iT−1.68e3T2 1 - 29.5iT - 1.68e3T^{2}
43 1−45.7T+1.84e3T2 1 - 45.7T + 1.84e3T^{2}
47 1−51.1iT−2.20e3T2 1 - 51.1iT - 2.20e3T^{2}
53 1+47.1iT−2.80e3T2 1 + 47.1iT - 2.80e3T^{2}
59 1+33.5iT−3.48e3T2 1 + 33.5iT - 3.48e3T^{2}
61 1+10.4T+3.72e3T2 1 + 10.4T + 3.72e3T^{2}
67 1−63.3T+4.48e3T2 1 - 63.3T + 4.48e3T^{2}
71 1+92.4iT−5.04e3T2 1 + 92.4iT - 5.04e3T^{2}
73 1+57.1T+5.32e3T2 1 + 57.1T + 5.32e3T^{2}
79 1+22.6T+6.24e3T2 1 + 22.6T + 6.24e3T^{2}
83 1−108.iT−6.88e3T2 1 - 108. iT - 6.88e3T^{2}
89 1−65.2iT−7.92e3T2 1 - 65.2iT - 7.92e3T^{2}
97 1+36.4T+9.40e3T2 1 + 36.4T + 9.40e3T^{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

−12.90023385097438979478076787096, −12.27924270596526933540493086876, −11.22599180745249722846961280159, −9.841043083665339362324136062802, −8.778553014256907857273566996704, −7.80329384800721536923613602227, −6.23303052282266849810451865403, −5.28595421453125009501401212438, −3.47904714578236888857879846136, −1.25861746030581259933347928955, 2.51536723491900765715101839783, 3.41984835846068180050624208098, 5.89623770846633214502606235923, 7.02805613860950361381779579016, 7.37772863586674204017816350766, 9.688292356225533676325210242785, 10.31550876049810390027514280728, 11.38838646361895986573295703718, 12.07385673838523335962015292042, 13.62830241114388327200715978027

Graph of the ZZ-function along the critical line