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Properties

Label 2-2156-44.31-c0-0-1
Degree 22
Conductor 21562156
Sign 0.0457+0.998i0.0457 + 0.998i
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 + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.809 + 0.587i)22-s − 1.17i·23-s + (0.809 − 0.587i)25-s + (0.5 + 0.363i)29-s − 32-s + (−0.809 − 0.587i)36-s + (−1.30 − 0.951i)37-s + 1.90i·43-s + 44-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (0.809 + 0.587i)22-s − 1.17i·23-s + (0.809 − 0.587i)25-s + (0.5 + 0.363i)29-s − 32-s + (−0.809 − 0.587i)36-s + (−1.30 − 0.951i)37-s + 1.90i·43-s + 44-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((0.0457+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s) L(s)=((0.0457+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 0.0457+0.998i0.0457 + 0.998i
Analytic conductor: 1.075981.07598
Root analytic conductor: 1.037291.03729
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2156(295,⋅)\chi_{2156} (295, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :0), 0.0457+0.998i)(2,\ 2156,\ (\ :0),\ 0.0457 + 0.998i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8829264961.882926496
L(12)L(\frac12) ≈\approx 1.8829264961.882926496
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+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
good3 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
5 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
13 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
17 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
19 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
23 1+1.17iT−T2 1 + 1.17iT - 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+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)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+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
67 1−1.90iT−T2 1 - 1.90iT - 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+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
79 1+(−1.80−0.587i)T+(0.809+0.587i)T2 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1+T2 1 + 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

−9.260619221646573952632310258993, −8.495189521230951502256498580448, −7.12300891015867356796205536148, −6.70132575449170887587267911925, −5.88619258511561184115702576009, −4.78564509680483174264859772263, −4.25746221996550414727774721627, −3.29622688538725096910354894141, −2.32198143447043050283840273335, −1.12812718897461128216231276572, 1.77272992327997460524891031736, 3.02559129580826860242237050292, 3.75375999522768963027245614725, 4.83291672661931920139357735444, 5.39930400202314447065443570226, 6.25604237199220609150750650274, 7.09382114724069700988116585290, 7.74241093801919321006829826139, 8.526718897051506495801929374660, 9.170270498135628545332788761732

Graph of the ZZ-function along the critical line