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Properties

Label 2-2156-308.207-c0-0-1
Degree 22
Conductor 21562156
Sign 0.440−0.897i0.440 - 0.897i
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)16-s + (0.5 − 0.866i)18-s + (−0.809 + 0.587i)22-s + (1.01 − 0.587i)23-s + (0.104 + 0.994i)25-s + (0.5 + 0.363i)29-s + (0.500 − 0.866i)32-s + (0.309 + 0.951i)36-s + (−0.169 + 1.60i)37-s − 1.90i·43-s + (0.104 − 0.994i)44-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)16-s + (0.5 − 0.866i)18-s + (−0.809 + 0.587i)22-s + (1.01 − 0.587i)23-s + (0.104 + 0.994i)25-s + (0.5 + 0.363i)29-s + (0.500 − 0.866i)32-s + (0.309 + 0.951i)36-s + (−0.169 + 1.60i)37-s − 1.90i·43-s + (0.104 − 0.994i)44-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((0.440−0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s) L(s)=((0.440−0.897i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 0.440−0.897i0.440 - 0.897i
Analytic conductor: 1.075981.07598
Root analytic conductor: 1.037291.03729
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2156(1439,⋅)\chi_{2156} (1439, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :0), 0.440−0.897i)(2,\ 2156,\ (\ :0),\ 0.440 - 0.897i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.80697239950.8069723995
L(12)L(\frac12) ≈\approx 0.80697239950.8069723995
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+(0.669−0.743i)T 1 + (0.669 - 0.743i)T
7 1 1
11 1+(−0.978−0.207i)T 1 + (-0.978 - 0.207i)T
good3 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
5 1+(−0.104−0.994i)T2 1 + (-0.104 - 0.994i)T^{2}
13 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
17 1+(0.913+0.406i)T2 1 + (0.913 + 0.406i)T^{2}
19 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
23 1+(−1.01+0.587i)T+(0.5−0.866i)T2 1 + (-1.01 + 0.587i)T + (0.5 - 0.866i)T^{2}
29 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
31 1+(0.104−0.994i)T2 1 + (0.104 - 0.994i)T^{2}
37 1+(0.169−1.60i)T+(−0.978−0.207i)T2 1 + (0.169 - 1.60i)T + (-0.978 - 0.207i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1+1.90iT−T2 1 + 1.90iT - T^{2}
47 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
53 1+(−0.413−0.459i)T+(−0.104+0.994i)T2 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2}
59 1+(−0.669+0.743i)T2 1 + (-0.669 + 0.743i)T^{2}
61 1+(−0.104−0.994i)T2 1 + (-0.104 - 0.994i)T^{2}
67 1+(−1.64−0.951i)T+(0.5+0.866i)T2 1 + (-1.64 - 0.951i)T + (0.5 + 0.866i)T^{2}
71 1+(−1.80+0.587i)T+(0.809−0.587i)T2 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2}
73 1+(0.669−0.743i)T2 1 + (0.669 - 0.743i)T^{2}
79 1+(−0.395−1.86i)T+(−0.913+0.406i)T2 1 + (-0.395 - 1.86i)T + (-0.913 + 0.406i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)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

−9.130028038160275433510137426693, −8.685942677687999537050725162291, −7.974482428991566736845477576649, −6.92055649974477032959923967532, −6.56991300137705390186050617789, −5.48219773625376493759171818070, −4.93703105365822843540017382220, −3.71095434632529169431390736870, −2.45755004255016039316357905292, −1.14768862277056526561571673262, 0.880608936908200303754094222642, 2.23007909578011927921338333246, 3.17529608929992371105153780895, 3.95166811278352345061858988313, 4.99512658484538926689602366344, 6.15128604865641851477438993031, 6.85764378650936362607384150821, 7.86902691816112529357822568261, 8.514509314307978002327384301147, 9.218746893590908631462904600639

Graph of the ZZ-function along the critical line