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Properties

Label 2-2156-44.15-c0-0-1
Degree 22
Conductor 21562156
Sign −0.286+0.958i-0.286 + 0.958i
Analytic cond. 1.075981.07598
Root an. cond. 1.037291.03729
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)16-s − 18-s + (0.309 − 0.951i)22-s − 1.90i·23-s + (−0.309 + 0.951i)25-s + (0.5 + 1.53i)29-s − 32-s + (−0.809 + 0.587i)36-s + (−0.190 − 0.587i)37-s − 1.17i·43-s + (−0.309 − 0.951i)44-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)16-s − 18-s + (0.309 − 0.951i)22-s − 1.90i·23-s + (−0.309 + 0.951i)25-s + (0.5 + 1.53i)29-s − 32-s + (−0.809 + 0.587i)36-s + (−0.190 − 0.587i)37-s − 1.17i·43-s + (−0.309 − 0.951i)44-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((−0.286+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s) L(s)=((−0.286+0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: −0.286+0.958i-0.286 + 0.958i
Analytic conductor: 1.075981.07598
Root analytic conductor: 1.037291.03729
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2156(1863,⋅)\chi_{2156} (1863, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :0), −0.286+0.958i)(2,\ 2156,\ (\ :0),\ -0.286 + 0.958i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7317147651.731714765
L(12)L(\frac12) ≈\approx 1.7317147651.731714765
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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
7 1 1
11 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
good3 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
5 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
13 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
17 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
19 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
23 1+1.90iT−T2 1 + 1.90iT - T^{2}
29 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2}
41 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
43 1+1.17iT−T2 1 + 1.17iT - T^{2}
47 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
53 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
59 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1+1.17iT−T2 1 + 1.17iT - T^{2}
71 1+(−0.690−0.951i)T+(−0.309+0.951i)T2 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(0.690−0.951i)T+(−0.309−0.951i)T2 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2}
83 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)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

−8.912864791482909603938348477830, −8.675237519144662839495920612250, −7.18127077204117080544613769631, −6.49033458759685869512369331743, −5.80168470568820513954718102093, −5.00792384787480750087675018877, −3.95570889537423960554649426246, −3.30080513058424071049616693781, −2.34378197902843575970533307881, −0.963113067085820361643528446419, 1.93613290028870277772552978196, 2.94963164383529625575826051288, 3.94747622880843381280010349217, 4.70402838962780319898764472890, 5.60841165878992448041848328950, 6.22730742995106082540599559250, 7.09280914247039021216855352188, 7.87671405816392816643041156320, 8.443798418431548599369426063837, 9.395294917626435868040980214970

Graph of the ZZ-function along the critical line