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Properties

Label 2-38e2-19.13-c0-0-2
Degree 22
Conductor 14441444
Sign 0.756+0.654i0.756 + 0.654i
Analytic cond. 0.7206490.720649
Root an. cond. 0.8489100.848910
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  + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)17-s + (0.347 − 1.96i)23-s + (−0.939 − 0.342i)35-s + (−0.173 − 0.984i)43-s + (0.5 − 0.866i)45-s + (−0.766 − 0.642i)47-s + (0.766 − 0.642i)55-s + (−0.173 + 0.984i)61-s + (0.939 − 0.342i)63-s + (0.939 + 0.342i)73-s + 0.999·77-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)5-s + (0.5 − 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)17-s + (0.347 − 1.96i)23-s + (−0.939 − 0.342i)35-s + (−0.173 − 0.984i)43-s + (0.5 − 0.866i)45-s + (−0.766 − 0.642i)47-s + (0.766 − 0.642i)55-s + (−0.173 + 0.984i)61-s + (0.939 − 0.342i)63-s + (0.939 + 0.342i)73-s + 0.999·77-s + ⋯

Functional equation

Λ(s)=(1444s/2 ΓC(s) L(s)=((0.756+0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1444s/2 ΓC(s) L(s)=((0.756+0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14441444    =    22⋅1922^{2} \cdot 19^{2}
Sign: 0.756+0.654i0.756 + 0.654i
Analytic conductor: 0.7206490.720649
Root analytic conductor: 0.8489100.848910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1444(849,⋅)\chi_{1444} (849, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1444, ( :0), 0.756+0.654i)(2,\ 1444,\ (\ :0),\ 0.756 + 0.654i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1938527651.193852765
L(12)L(\frac12) ≈\approx 1.1938527651.193852765
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)
bad2 1 1
19 1 1
good3 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
5 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
7 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
17 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
23 1+(−0.347+1.96i)T+(−0.939−0.342i)T2 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2}
29 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
43 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
47 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}
53 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
59 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
61 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
67 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
71 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
73 1+(−0.939−0.342i)T+(0.766+0.642i)T2 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}
79 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
83 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
89 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
97 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)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

−9.657578550575018497446384065862, −8.667460430270846566343263709993, −8.166662536804208817896250103342, −7.15939018981629681069160374730, −6.61857664787464691611245880116, −5.16782434056809346188552814760, −4.44584563969055642202848499214, −4.08591757090576855183888262331, −2.22966226946056225556464826614, −1.18261387442914511522637451506, 1.56033483914506031062019341689, 2.90531903490272118231621679504, 3.63255962710652532149526077022, 4.79951486384512001158566651136, 5.81411447069883320557557037233, 6.61480820465365939561459036228, 7.28416328862524800532997119226, 8.201256778679885632052242881099, 9.177587320676378765366627570587, 9.602402202595463281478134415015

Graph of the ZZ-function along the critical line