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Properties

Label 2-1368-152.75-c1-0-93
Degree 22
Conductor 13681368
Sign −i-i
Analytic cond. 10.923510.9235
Root an. cond. 3.305073.30507
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s − 4.32i·5-s + 2.82i·8-s − 6.11·10-s − 4.15·13-s + 4.00·16-s − 4.35·19-s + 8.65i·20-s − 1.55i·23-s − 13.7·25-s + 5.87i·26-s + 10.2·31-s − 5.65i·32-s − 8.07·37-s + 6.16i·38-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 1.93i·5-s + 1.00i·8-s − 1.93·10-s − 1.15·13-s + 1.00·16-s − 1.00·19-s + 1.93i·20-s − 0.323i·23-s − 2.74·25-s + 1.15i·26-s + 1.84·31-s − 1.00i·32-s − 1.32·37-s + 1.00i·38-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: −i-i
Analytic conductor: 10.923510.9235
Root analytic conductor: 3.305073.30507
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1368(379,⋅)\chi_{1368} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :1/2), −i)(2,\ 1368,\ (\ :1/2),\ -i)

Particular Values

L(1)L(1) ≈\approx 0.45625195960.4562519596
L(12)L(\frac12) ≈\approx 0.45625195960.4562519596
L(32)L(\frac{3}{2}) 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+1.41iT 1 + 1.41iT
3 1 1
19 1+4.35T 1 + 4.35T
good5 1+4.32iT−5T2 1 + 4.32iT - 5T^{2}
7 1−7T2 1 - 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+4.15T+13T2 1 + 4.15T + 13T^{2}
17 1+17T2 1 + 17T^{2}
23 1+1.55iT−23T2 1 + 1.55iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−10.2T+31T2 1 - 10.2T + 31T^{2}
37 1+8.07T+37T2 1 + 8.07T + 37T^{2}
41 1−12.3iT−41T2 1 - 12.3iT - 41T^{2}
43 1+8.71T+43T2 1 + 8.71T + 43T^{2}
47 1+7.10iT−47T2 1 + 7.10iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+2.82iT−59T2 1 + 2.82iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+8.71T+73T2 1 + 8.71T + 73T^{2}
79 1−14.1T+79T2 1 - 14.1T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+11.3iT−89T2 1 + 11.3iT - 89T^{2}
97 1−97T2 1 - 97T^{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

−8.980887530064943296208838819984, −8.484378516264037348432768644731, −7.80604097105995035548927458280, −6.31095464701282956399213403489, −5.02270909955010252979871011808, −4.83325807295692173920753371230, −3.87857896447073635315498801478, −2.46066577083743670762932046404, −1.40176069244099492316005153129, −0.18890245370169336375534339194, 2.31291787234767368585972327546, 3.32685850671220338687995147555, 4.30530776517581053640086842033, 5.43051921842333032957900690151, 6.37740478575496866121812422003, 6.90655042028833515328307090098, 7.51003541078299752915522088937, 8.341281833193170729711527303137, 9.398275790711519551742976708899, 10.29406552428233469191472194390

Graph of the ZZ-function along the critical line