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Properties

Label 2-1368-1368.371-c0-0-1
Degree 22
Conductor 13681368
Sign 0.150−0.988i0.150 - 0.988i
Analytic cond. 0.6827200.682720
Root an. cond. 0.8262690.826269
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  + (0.939 + 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s − 6-s + (0.500 + 0.866i)8-s + (0.766 − 0.642i)9-s + 0.684i·11-s + (−0.939 − 0.342i)12-s + (0.173 + 0.984i)16-s + (0.939 − 0.342i)18-s + (−0.173 + 0.984i)19-s + (−0.233 + 0.642i)22-s + (−0.766 − 0.642i)24-s + (0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s − 6-s + (0.500 + 0.866i)8-s + (0.766 − 0.642i)9-s + 0.684i·11-s + (−0.939 − 0.342i)12-s + (0.173 + 0.984i)16-s + (0.939 − 0.342i)18-s + (−0.173 + 0.984i)19-s + (−0.233 + 0.642i)22-s + (−0.766 − 0.642i)24-s + (0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + ⋯

Functional equation

Λ(s)=(1368s/2 ΓC(s) L(s)=((0.150−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1368s/2 ΓC(s) L(s)=((0.150−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: 0.150−0.988i0.150 - 0.988i
Analytic conductor: 0.6827200.682720
Root analytic conductor: 0.8262690.826269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1368(371,⋅)\chi_{1368} (371, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :0), 0.150−0.988i)(2,\ 1368,\ (\ :0),\ 0.150 - 0.988i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4423844221.442384422
L(12)L(\frac12) ≈\approx 1.4423844221.442384422
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+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
3 1+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
19 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
good5 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1−0.684iT−T2 1 - 0.684iT - T^{2}
13 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
17 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
23 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
29 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
43 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
47 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
53 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
59 1+(0.266−0.223i)T+(0.173−0.984i)T2 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2}
61 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
67 1+(0.439+1.20i)T+(−0.766+0.642i)T2 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(−1.43+1.20i)T+(0.173−0.984i)T2 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2}
79 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
83 1+(0.592−0.342i)T+(0.5−0.866i)T2 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2}
89 1+(−0.766−0.642i)T+(0.173+0.984i)T2 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}
97 1+(−0.673+1.85i)T+(−0.766−0.642i)T2 1 + (-0.673 + 1.85i)T + (-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

−10.27880514618393272207230992769, −9.234972554787684811174412518335, −8.126119639914055054434251852605, −7.21051533946065495669638714581, −6.52783148661944558154857315722, −5.75280395955028449468856491017, −4.93296328195058961177954436037, −4.24756089958659077891570123452, −3.27826402835457662213222772579, −1.80129896114823085287066034617, 1.10350110481387059667592179699, 2.45604023922574924976838689234, 3.59545070219792478914290095412, 4.74557935608395493673151184842, 5.28047599475956335546549378422, 6.27098791822153467169276984167, 6.78782047437833212056569685717, 7.68772870993097268926440412529, 8.859912192894589656635933497594, 9.970031995357624874598958487674

Graph of the ZZ-function along the critical line