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Properties

Label 2-2790-5.4-c1-0-54
Degree 22
Conductor 27902790
Sign 0.0235+0.999i0.0235 + 0.999i
Analytic cond. 22.278222.2782
Root an. cond. 4.719984.71998
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  − i·2-s − 4-s + (2.23 − 0.0526i)5-s + 2i·7-s + i·8-s + (−0.0526 − 2.23i)10-s + 0.470·11-s − 6.47i·13-s + 2·14-s + 16-s − 7.04i·17-s + 7.04·19-s + (−2.23 + 0.0526i)20-s − 0.470i·22-s + 6.94i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.999 − 0.0235i)5-s + 0.755i·7-s + 0.353i·8-s + (−0.0166 − 0.706i)10-s + 0.141·11-s − 1.79i·13-s + 0.534·14-s + 0.250·16-s − 1.70i·17-s + 1.61·19-s + (−0.499 + 0.0117i)20-s − 0.100i·22-s + 1.44i·23-s + ⋯

Functional equation

Λ(s)=(2790s/2 ΓC(s) L(s)=((0.0235+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2790s/2 ΓC(s+1/2) L(s)=((0.0235+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27902790    =    2⋅32⋅5⋅312 \cdot 3^{2} \cdot 5 \cdot 31
Sign: 0.0235+0.999i0.0235 + 0.999i
Analytic conductor: 22.278222.2782
Root analytic conductor: 4.719984.71998
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2790(559,⋅)\chi_{2790} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2790, ( :1/2), 0.0235+0.999i)(2,\ 2790,\ (\ :1/2),\ 0.0235 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 2.1066401242.106640124
L(12)L(\frac12) ≈\approx 2.1066401242.106640124
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+iT 1 + iT
3 1 1
5 1+(−2.23+0.0526i)T 1 + (-2.23 + 0.0526i)T
31 1+T 1 + T
good7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1−0.470T+11T2 1 - 0.470T + 11T^{2}
13 1+6.47iT−13T2 1 + 6.47iT - 13T^{2}
17 1+7.04iT−17T2 1 + 7.04iT - 17T^{2}
19 1−7.04T+19T2 1 - 7.04T + 19T^{2}
23 1−6.94iT−23T2 1 - 6.94iT - 23T^{2}
29 1+6.94T+29T2 1 + 6.94T + 29T^{2}
37 1+1.78iT−37T2 1 + 1.78iT - 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−0.210iT−43T2 1 - 0.210iT - 43T^{2}
47 1−7.04iT−47T2 1 - 7.04iT - 47T^{2}
53 1+3.15iT−53T2 1 + 3.15iT - 53T^{2}
59 1+7.15T+59T2 1 + 7.15T + 59T^{2}
61 1−11.2T+61T2 1 - 11.2T + 61T^{2}
67 1+8.26iT−67T2 1 + 8.26iT - 67T^{2}
71 1+0.260T+71T2 1 + 0.260T + 71T^{2}
73 1+11.8iT−73T2 1 + 11.8iT - 73T^{2}
79 1−1.89T+79T2 1 - 1.89T + 79T^{2}
83 1+11.7iT−83T2 1 + 11.7iT - 83T^{2}
89 1−12.0T+89T2 1 - 12.0T + 89T^{2}
97 1+3.52iT−97T2 1 + 3.52iT - 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

−9.050571595616411472124151095253, −7.81849163980282133141721908362, −7.32447665526712799931346977387, −5.96807425500609589463371179240, −5.37326059016530776534817297812, −5.03222681457935827246980558803, −3.35340912534452124701244545190, −2.94367296023099557841837068034, −1.89776106493156159663114676105, −0.75792939333163638015589463304, 1.21612246439137932815143577702, 2.16351428838903231664165113571, 3.65404957519421756327404047465, 4.30609868242119966551790822875, 5.26765270771681035386536696096, 6.04463519627878470491845642003, 6.76132400941566487411732425576, 7.21289214385532651497717921011, 8.285494449632933689188766654107, 8.964770598895627632474784165002

Graph of the ZZ-function along the critical line