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Properties

Label 2-345-345.29-c0-0-0
Degree 22
Conductor 345345
Sign 0.392−0.919i0.392 - 0.919i
Analytic cond. 0.1721770.172177
Root an. cond. 0.4149420.414942
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  + (−0.797 + 1.74i)2-s + (−0.959 − 0.281i)3-s + (−1.75 − 2.02i)4-s + (0.841 − 0.540i)5-s + (1.25 − 1.45i)6-s + (3.09 − 0.909i)8-s + (0.841 + 0.540i)9-s + (0.273 + 1.89i)10-s + (1.11 + 2.44i)12-s + (−0.959 + 0.281i)15-s + (−0.500 + 3.47i)16-s + (0.857 − 0.989i)17-s + (−1.61 + 1.03i)18-s + (0.186 + 0.215i)19-s + (−2.57 − 0.755i)20-s + ⋯
L(s)  = 1  + (−0.797 + 1.74i)2-s + (−0.959 − 0.281i)3-s + (−1.75 − 2.02i)4-s + (0.841 − 0.540i)5-s + (1.25 − 1.45i)6-s + (3.09 − 0.909i)8-s + (0.841 + 0.540i)9-s + (0.273 + 1.89i)10-s + (1.11 + 2.44i)12-s + (−0.959 + 0.281i)15-s + (−0.500 + 3.47i)16-s + (0.857 − 0.989i)17-s + (−1.61 + 1.03i)18-s + (0.186 + 0.215i)19-s + (−2.57 − 0.755i)20-s + ⋯

Functional equation

Λ(s)=(345s/2 ΓC(s) L(s)=((0.392−0.919i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(345s/2 ΓC(s) L(s)=((0.392−0.919i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 345345    =    3⋅5⋅233 \cdot 5 \cdot 23
Sign: 0.392−0.919i0.392 - 0.919i
Analytic conductor: 0.1721770.172177
Root analytic conductor: 0.4149420.414942
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ345(29,⋅)\chi_{345} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 345, ( :0), 0.392−0.919i)(2,\ 345,\ (\ :0),\ 0.392 - 0.919i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.44419619620.4441961962
L(12)L(\frac12) ≈\approx 0.44419619620.4441961962
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+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
5 1+(−0.841+0.540i)T 1 + (-0.841 + 0.540i)T
23 1+(−0.415−0.909i)T 1 + (-0.415 - 0.909i)T
good2 1+(0.797−1.74i)T+(−0.654−0.755i)T2 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2}
7 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
11 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
13 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
17 1+(−0.857+0.989i)T+(−0.142−0.989i)T2 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2}
19 1+(−0.186−0.215i)T+(−0.142+0.989i)T2 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2}
29 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
31 1+(−0.273+0.0801i)T+(0.841−0.540i)T2 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2}
37 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
41 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
43 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
47 1−0.830T+T2 1 - 0.830T + T^{2}
53 1+(0.118−0.822i)T+(−0.959−0.281i)T2 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2}
59 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
61 1+(0.797−0.234i)T+(0.841−0.540i)T2 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2}
67 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
71 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
73 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
79 1+(0.239+1.66i)T+(−0.959+0.281i)T2 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2}
83 1+(1.61+1.03i)T+(0.415+0.909i)T2 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2}
89 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
97 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)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

−11.91089065759182426988354114824, −10.54961406989724961272142724406, −9.764595797492933284693129556603, −9.096118387094815778649703111029, −7.87399618298587003716072794061, −7.12876864929447214911294639144, −6.11542446581340811521417616236, −5.46272047483734989227850091498, −4.70853200876993286001993447405, −1.23386469457785985176323931721, 1.46459226336072092376792273284, 2.93522196579340408021302567228, 4.17102270130317430548406875996, 5.45391537050398861032702333170, 6.81838041435685354096406553739, 8.188313847907576118045191991167, 9.366145756786886366382605886330, 10.00029626495290873643209817560, 10.68446215545145526478523557791, 11.21942454144155133550407183043

Graph of the ZZ-function along the critical line