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Properties

Label 2-3645-135.59-c0-0-3
Degree 22
Conductor 36453645
Sign 0.727−0.686i0.727 - 0.686i
Analytic cond. 1.819091.81909
Root an. cond. 1.348731.34873
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43 + 1.20i)2-s + (0.439 − 2.49i)4-s + (−0.939 + 0.342i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (−2.70 − 0.984i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (0.439 + 2.49i)20-s + (0.266 − 1.50i)23-s + (0.766 − 0.642i)25-s + (−0.326 + 1.85i)31-s + (2.37 − 0.866i)32-s + (−0.113 − 0.642i)34-s + (0.613 + 0.223i)38-s + ⋯
L(s)  = 1  + (−1.43 + 1.20i)2-s + (0.439 − 2.49i)4-s + (−0.939 + 0.342i)5-s + (1.43 + 2.49i)8-s + (0.939 − 1.62i)10-s + (−2.70 − 0.984i)16-s + (−0.173 + 0.300i)17-s + (−0.173 − 0.300i)19-s + (0.439 + 2.49i)20-s + (0.266 − 1.50i)23-s + (0.766 − 0.642i)25-s + (−0.326 + 1.85i)31-s + (2.37 − 0.866i)32-s + (−0.113 − 0.642i)34-s + (0.613 + 0.223i)38-s + ⋯

Functional equation

Λ(s)=(3645s/2 ΓC(s) L(s)=((0.727−0.686i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3645s/2 ΓC(s) L(s)=((0.727−0.686i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36453645    =    36⋅53^{6} \cdot 5
Sign: 0.727−0.686i0.727 - 0.686i
Analytic conductor: 1.819091.81909
Root analytic conductor: 1.348731.34873
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3645(404,⋅)\chi_{3645} (404, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3645, ( :0), 0.727−0.686i)(2,\ 3645,\ (\ :0),\ 0.727 - 0.686i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.42567265980.4256726598
L(12)L(\frac12) ≈\approx 0.42567265980.4256726598
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 1
5 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
good2 1+(1.43−1.20i)T+(0.173−0.984i)T2 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2}
7 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
11 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
13 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
17 1+(0.173−0.300i)T+(−0.5−0.866i)T2 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2}
19 1+(0.173+0.300i)T+(−0.5+0.866i)T2 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.266+1.50i)T+(−0.939−0.342i)T2 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(0.326−1.85i)T+(−0.939−0.342i)T2 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
43 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
47 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
53 1−1.53T+T2 1 - 1.53T + T^{2}
59 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
61 1+(0.326+1.85i)T+(−0.939+0.342i)T2 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2}
67 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−1.17+0.984i)T+(0.173−0.984i)T2 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2}
83 1+(−0.266+0.223i)T+(0.173−0.984i)T2 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)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

−8.577347065091471986556174451238, −8.188633236726796233706539989438, −7.35850767033167572505889820119, −6.77122457426919923074650227200, −6.33311922646595650848629177910, −5.21588890274289728065229741595, −4.52451584254735246982064114925, −3.25482152994149404200979986188, −1.97415503454697068282505049397, −0.59216053786114074988228852767, 0.818411285232987218250108680304, 1.89360636568325936514760275043, 2.92490175828837020140944700301, 3.72524283649598164097925075182, 4.36836040833147650588306556614, 5.60522124914372545311094493305, 6.90367342704278733785075911700, 7.60953488240072512641321024474, 7.992594637785512745229878805060, 8.800201893510720247878880929981

Graph of the ZZ-function along the critical line