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Properties

Label 2-1323-49.38-c0-0-0
Degree 22
Conductor 13231323
Sign 0.180−0.983i0.180 - 0.983i
Analytic cond. 0.6602630.660263
Root an. cond. 0.8125650.812565
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  + (−0.0747 + 0.997i)4-s + (0.222 + 0.974i)7-s + (1.81 − 0.414i)13-s + (−0.988 − 0.149i)16-s + (−1.68 + 0.974i)19-s + (−0.733 − 0.680i)25-s + (−0.988 + 0.149i)28-s + (1.72 + 0.997i)31-s + (−0.0111 − 0.149i)37-s + (−1.19 + 1.49i)43-s + (−0.900 + 0.433i)49-s + (0.277 + 1.84i)52-s + (1.12 − 0.0841i)61-s + (0.222 − 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)4-s + (0.222 + 0.974i)7-s + (1.81 − 0.414i)13-s + (−0.988 − 0.149i)16-s + (−1.68 + 0.974i)19-s + (−0.733 − 0.680i)25-s + (−0.988 + 0.149i)28-s + (1.72 + 0.997i)31-s + (−0.0111 − 0.149i)37-s + (−1.19 + 1.49i)43-s + (−0.900 + 0.433i)49-s + (0.277 + 1.84i)52-s + (1.12 − 0.0841i)61-s + (0.222 − 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.180−0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s) L(s)=((0.180−0.983i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.180−0.983i0.180 - 0.983i
Analytic conductor: 0.6602630.660263
Root analytic conductor: 0.8125650.812565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1323(136,⋅)\chi_{1323} (136, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :0), 0.180−0.983i)(2,\ 1323,\ (\ :0),\ 0.180 - 0.983i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0758468651.075846865
L(12)L(\frac12) ≈\approx 1.0758468651.075846865
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
7 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
good2 1+(0.0747−0.997i)T2 1 + (0.0747 - 0.997i)T^{2}
5 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
11 1+(0.826+0.563i)T2 1 + (0.826 + 0.563i)T^{2}
13 1+(−1.81+0.414i)T+(0.900−0.433i)T2 1 + (-1.81 + 0.414i)T + (0.900 - 0.433i)T^{2}
17 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
19 1+(1.68−0.974i)T+(0.5−0.866i)T2 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2}
23 1+(0.365+0.930i)T2 1 + (0.365 + 0.930i)T^{2}
29 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
31 1+(−1.72−0.997i)T+(0.5+0.866i)T2 1 + (-1.72 - 0.997i)T + (0.5 + 0.866i)T^{2}
37 1+(0.0111+0.149i)T+(−0.988+0.149i)T2 1 + (0.0111 + 0.149i)T + (-0.988 + 0.149i)T^{2}
41 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
43 1+(1.19−1.49i)T+(−0.222−0.974i)T2 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2}
47 1+(−0.0747+0.997i)T2 1 + (-0.0747 + 0.997i)T^{2}
53 1+(−0.988−0.149i)T2 1 + (-0.988 - 0.149i)T^{2}
59 1+(0.733−0.680i)T2 1 + (0.733 - 0.680i)T^{2}
61 1+(−1.12+0.0841i)T+(0.988−0.149i)T2 1 + (-1.12 + 0.0841i)T + (0.988 - 0.149i)T^{2}
67 1+(−0.988+1.71i)T+(−0.5−0.866i)T2 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2}
71 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
73 1+(−0.590+0.636i)T+(−0.0747−0.997i)T2 1 + (-0.590 + 0.636i)T + (-0.0747 - 0.997i)T^{2}
79 1+(0.826+1.43i)T+(−0.5+0.866i)T2 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
89 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
97 1+0.589iT−T2 1 + 0.589iT - 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

−9.998529549131960381130610885977, −8.862232654779607006235516407870, −8.291530304148654506061782976767, −8.035879818481224348354495169923, −6.45848012398651784117420106484, −6.14287271740877729442392936564, −4.82243758308665854434312827617, −3.88735371670369938120775058021, −3.01902831265171959151218343778, −1.85379443832162937236185699105, 0.991137069325056654341010841400, 2.15330192963795843913373422387, 3.83385641918393468580019432618, 4.42153385009983797655690119676, 5.52638943562908967838367873167, 6.46123214152476039422857128174, 6.88991208943601164591494960678, 8.257598920143090784653448289670, 8.761462467976784143889616795126, 9.828507970679896327810118952640

Graph of the ZZ-function along the critical line