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Properties

Label 2-1368-1368.1075-c0-0-0
Degree 22
Conductor 13681368
Sign −0.790−0.612i-0.790 - 0.612i
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  + i·2-s − 3-s − 4-s + (−0.866 + 0.5i)5-s − i·6-s + (0.866 − 0.5i)7-s − i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.5 + 0.866i)14-s + (0.866 − 0.5i)15-s + 16-s + (−0.5 + 0.866i)17-s + i·18-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s + (−0.866 + 0.5i)5-s − i·6-s + (0.866 − 0.5i)7-s − i·8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 12-s + (0.5 + 0.866i)14-s + (0.866 − 0.5i)15-s + 16-s + (−0.5 + 0.866i)17-s + i·18-s + ⋯

Functional equation

Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.790−0.612i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.790−0.612i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: −0.790−0.612i-0.790 - 0.612i
Analytic conductor: 0.6827200.682720
Root analytic conductor: 0.8262690.826269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1368(1075,⋅)\chi_{1368} (1075, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :0), −0.790−0.612i)(2,\ 1368,\ (\ :0),\ -0.790 - 0.612i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.59279222880.5927922288
L(12)L(\frac12) ≈\approx 0.59279222880.5927922288
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−iT 1 - iT
3 1+T 1 + T
19 1−T 1 - T
good5 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
7 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)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−T2 1 - T^{2}
17 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1−T2 1 - T^{2}
29 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
31 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
37 1−2iT−T2 1 - 2iT - T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+T2 1 + T^{2}
47 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
67 1−2T+T2 1 - 2T + T^{2}
71 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1−T2 1 - T^{2}
83 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
89 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+T2 1 + 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

−9.997845002965037758416562084212, −9.357892879685720733896280210297, −7.996315755544614159341528969600, −7.61761555947485876040635225999, −6.88338426545065939874180397311, −6.14046629083461239473195480549, −5.08414426631715580775427205765, −4.37402831368063362271674168739, −3.70736182725343264273357302482, −1.41040769341772955298021833895, 0.64883884252614936288081504103, 1.93328793093652259146125002360, 3.50908298697910065660798941257, 4.30356943788939694742619366100, 5.20759666615194740125820140503, 5.68533425837039125906128231297, 7.18651228174847546650366281945, 7.950783588858228305544409918211, 8.969078739805931645637729353509, 9.353326675013679398010100654638

Graph of the ZZ-function along the critical line