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Properties

Label 2-1368-1368.1139-c0-0-0
Degree 22
Conductor 13681368
Sign −0.642−0.766i-0.642 - 0.766i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + 1.73i·17-s − 0.999·18-s + (0.5 + 0.866i)19-s + (−1.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 + 0.866i)11-s + 0.999·12-s + (−0.5 − 0.866i)16-s + 1.73i·17-s − 0.999·18-s + (0.5 + 0.866i)19-s + (−1.5 − 0.866i)22-s + (0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: −0.642−0.766i-0.642 - 0.766i
Analytic conductor: 0.6827200.682720
Root analytic conductor: 0.8262690.826269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1368(1139,⋅)\chi_{1368} (1139, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :0), −0.642−0.766i)(2,\ 1368,\ (\ :0),\ -0.642 - 0.766i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.78013584600.7801358460
L(12)L(\frac12) ≈\approx 0.78013584600.7801358460
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
13 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
17 1−1.73iT−T2 1 - 1.73iT - T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1−T2 1 - 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+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T+T2 1 - T + T^{2}
79 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+2T+T2 1 + 2T + T^{2}
97 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)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

−10.19992407014213507329314893543, −8.953728124224133310576321510780, −8.002279276761307831718260393146, −7.63670378739673953381509578231, −6.83068697769412381049165946727, −5.86494250506469358928852376920, −5.38045687324001764086945309329, −4.40955263132450233428023203095, −3.14820470194036311902705481718, −1.88196372475311117823275748292, 0.57279851338610339336732845235, 2.73257050027175518378664748740, 3.15647844675266913186141739174, 4.56571333109753388850040806957, 5.04881230319214055912586177595, 5.77333008573524034871407430954, 6.78924731069484555945057345848, 8.110951552174974471932237545516, 9.043044991540750818265817362150, 9.742837198637934637974034440040

Graph of the ZZ-function along the critical line