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Properties

Label 2-117-39.38-c2-0-5
Degree 22
Conductor 117117
Sign 0.999−0.0175i0.999 - 0.0175i
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  + 3.11·2-s + 5.69·4-s + 2.14·5-s + 1.36i·7-s + 5.26·8-s + 6.69·10-s + 0.221·11-s + (−7.69 − 10.4i)13-s + 4.24i·14-s − 6.38·16-s + 24.1i·17-s − 27.7i·19-s + 12.2·20-s + 0.690·22-s + 28.3i·23-s + ⋯
L(s)  = 1  + 1.55·2-s + 1.42·4-s + 0.429·5-s + 0.194i·7-s + 0.657·8-s + 0.669·10-s + 0.0201·11-s + (−0.591 − 0.806i)13-s + 0.303i·14-s − 0.398·16-s + 1.42i·17-s − 1.46i·19-s + 0.611·20-s + 0.0313·22-s + 1.23i·23-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.999−0.0175i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0175i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.999−0.0175i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.999−0.0175i0.999 - 0.0175i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(116,⋅)\chi_{117} (116, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), 0.999−0.0175i)(2,\ 117,\ (\ :1),\ 0.999 - 0.0175i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.98805+0.0261843i2.98805 + 0.0261843i
L(12)L(\frac12) ≈\approx 2.98805+0.0261843i2.98805 + 0.0261843i
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+(7.69+10.4i)T 1 + (7.69 + 10.4i)T
good2 1−3.11T+4T2 1 - 3.11T + 4T^{2}
5 1−2.14T+25T2 1 - 2.14T + 25T^{2}
7 1−1.36iT−49T2 1 - 1.36iT - 49T^{2}
11 1−0.221T+121T2 1 - 0.221T + 121T^{2}
17 1−24.1iT−289T2 1 - 24.1iT - 289T^{2}
19 1+27.7iT−361T2 1 + 27.7iT - 361T^{2}
23 1−28.3iT−529T2 1 - 28.3iT - 529T^{2}
29 1+27.0iT−841T2 1 + 27.0iT - 841T^{2}
31 1−18.6iT−961T2 1 - 18.6iT - 961T^{2}
37 1+20.0iT−1.36e3T2 1 + 20.0iT - 1.36e3T^{2}
41 1+14.6T+1.68e3T2 1 + 14.6T + 1.68e3T^{2}
43 1−30.1T+1.84e3T2 1 - 30.1T + 1.84e3T^{2}
47 1−78.7T+2.20e3T2 1 - 78.7T + 2.20e3T^{2}
53 1−26.7iT−2.80e3T2 1 - 26.7iT - 2.80e3T^{2}
59 1−52.3T+3.48e3T2 1 - 52.3T + 3.48e3T^{2}
61 1+11.2T+3.72e3T2 1 + 11.2T + 3.72e3T^{2}
67 1+76.0iT−4.48e3T2 1 + 76.0iT - 4.48e3T^{2}
71 1+75.0T+5.04e3T2 1 + 75.0T + 5.04e3T^{2}
73 1−122.iT−5.32e3T2 1 - 122. iT - 5.32e3T^{2}
79 1−62.6T+6.24e3T2 1 - 62.6T + 6.24e3T^{2}
83 1−94.9T+6.88e3T2 1 - 94.9T + 6.88e3T^{2}
89 1−21.4T+7.92e3T2 1 - 21.4T + 7.92e3T^{2}
97 1+133.iT−9.40e3T2 1 + 133. iT - 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

−13.31877583908627315468010619950, −12.56523624105233828567446465600, −11.58666457941826692669616063875, −10.44145265362339689744019777533, −9.081841383559654982407255426569, −7.46598879426445458531261358213, −6.09753907390752052988824869088, −5.27871167316522094180107909537, −3.91255435858274359424700093379, −2.44814460435717312872983340000, 2.40645258341439104543050965701, 3.98391998312522841863972510343, 5.12256823743093219097407264140, 6.24847405864604929947245678453, 7.36547685809360030384950955445, 9.135039432155504654463147018735, 10.36241275898755473043373895081, 11.73363893406068270662478091773, 12.32661153554343913745469682390, 13.50106410878804419539122704473

Graph of the ZZ-function along the critical line