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Properties

Label 2-92-23.2-c1-0-1
Degree 22
Conductor 9292
Sign −0.00508+0.999i-0.00508 + 0.999i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
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  + (−0.396 − 2.75i)3-s + (−1.50 − 0.441i)5-s + (−0.107 + 0.124i)7-s + (−4.57 + 1.34i)9-s + (5.36 − 3.44i)11-s + (2.99 + 3.45i)13-s + (−0.621 + 4.32i)15-s + (−0.232 + 0.510i)17-s + (1.75 + 3.85i)19-s + (0.386 + 0.248i)21-s + (−2.40 − 4.14i)23-s + (−2.13 − 1.37i)25-s + (2.04 + 4.47i)27-s + (−2.34 + 5.12i)29-s + (−1.27 + 8.86i)31-s + ⋯
L(s)  = 1  + (−0.228 − 1.59i)3-s + (−0.672 − 0.197i)5-s + (−0.0407 + 0.0470i)7-s + (−1.52 + 0.447i)9-s + (1.61 − 1.03i)11-s + (0.829 + 0.957i)13-s + (−0.160 + 1.11i)15-s + (−0.0565 + 0.123i)17-s + (0.403 + 0.883i)19-s + (0.0843 + 0.0541i)21-s + (−0.501 − 0.865i)23-s + (−0.427 − 0.274i)25-s + (0.393 + 0.861i)27-s + (−0.434 + 0.951i)29-s + (−0.228 + 1.59i)31-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((−0.00508+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00508 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+1/2) L(s)=((−0.00508+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00508 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: −0.00508+0.999i-0.00508 + 0.999i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ92(25,⋅)\chi_{92} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :1/2), −0.00508+0.999i)(2,\ 92,\ (\ :1/2),\ -0.00508 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.617620−0.620767i0.617620 - 0.620767i
L(12)L(\frac12) ≈\approx 0.617620−0.620767i0.617620 - 0.620767i
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 1
23 1+(2.40+4.14i)T 1 + (2.40 + 4.14i)T
good3 1+(0.396+2.75i)T+(−2.87+0.845i)T2 1 + (0.396 + 2.75i)T + (-2.87 + 0.845i)T^{2}
5 1+(1.50+0.441i)T+(4.20+2.70i)T2 1 + (1.50 + 0.441i)T + (4.20 + 2.70i)T^{2}
7 1+(0.107−0.124i)T+(−0.996−6.92i)T2 1 + (0.107 - 0.124i)T + (-0.996 - 6.92i)T^{2}
11 1+(−5.36+3.44i)T+(4.56−10.0i)T2 1 + (-5.36 + 3.44i)T + (4.56 - 10.0i)T^{2}
13 1+(−2.99−3.45i)T+(−1.85+12.8i)T2 1 + (-2.99 - 3.45i)T + (-1.85 + 12.8i)T^{2}
17 1+(0.232−0.510i)T+(−11.1−12.8i)T2 1 + (0.232 - 0.510i)T + (-11.1 - 12.8i)T^{2}
19 1+(−1.75−3.85i)T+(−12.4+14.3i)T2 1 + (-1.75 - 3.85i)T + (-12.4 + 14.3i)T^{2}
29 1+(2.34−5.12i)T+(−18.9−21.9i)T2 1 + (2.34 - 5.12i)T + (-18.9 - 21.9i)T^{2}
31 1+(1.27−8.86i)T+(−29.7−8.73i)T2 1 + (1.27 - 8.86i)T + (-29.7 - 8.73i)T^{2}
37 1+(−5.69+1.67i)T+(31.1−20.0i)T2 1 + (-5.69 + 1.67i)T + (31.1 - 20.0i)T^{2}
41 1+(−3.37−0.991i)T+(34.4+22.1i)T2 1 + (-3.37 - 0.991i)T + (34.4 + 22.1i)T^{2}
43 1+(1.17+8.15i)T+(−41.2+12.1i)T2 1 + (1.17 + 8.15i)T + (-41.2 + 12.1i)T^{2}
47 1+5.52T+47T2 1 + 5.52T + 47T^{2}
53 1+(−1.22+1.41i)T+(−7.54−52.4i)T2 1 + (-1.22 + 1.41i)T + (-7.54 - 52.4i)T^{2}
59 1+(4.03+4.65i)T+(−8.39+58.3i)T2 1 + (4.03 + 4.65i)T + (-8.39 + 58.3i)T^{2}
61 1+(0.202−1.40i)T+(−58.5−17.1i)T2 1 + (0.202 - 1.40i)T + (-58.5 - 17.1i)T^{2}
67 1+(6.36+4.09i)T+(27.8+60.9i)T2 1 + (6.36 + 4.09i)T + (27.8 + 60.9i)T^{2}
71 1+(1.68+1.08i)T+(29.4+64.5i)T2 1 + (1.68 + 1.08i)T + (29.4 + 64.5i)T^{2}
73 1+(−4.50−9.85i)T+(−47.8+55.1i)T2 1 + (-4.50 - 9.85i)T + (-47.8 + 55.1i)T^{2}
79 1+(−5.70−6.58i)T+(−11.2+78.1i)T2 1 + (-5.70 - 6.58i)T + (-11.2 + 78.1i)T^{2}
83 1+(−0.778+0.228i)T+(69.8−44.8i)T2 1 + (-0.778 + 0.228i)T + (69.8 - 44.8i)T^{2}
89 1+(−0.923−6.42i)T+(−85.3+25.0i)T2 1 + (-0.923 - 6.42i)T + (-85.3 + 25.0i)T^{2}
97 1+(7.25+2.13i)T+(81.6+52.4i)T2 1 + (7.25 + 2.13i)T + (81.6 + 52.4i)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

−13.87719828237322816234367908278, −12.52266066275554408374720342237, −11.89546949185866168717389197394, −11.08370732871731976402166823692, −8.979653175477284962532040886195, −8.121981335637259796114686103481, −6.81681128465316985589740164654, −6.05276335612152903536262310375, −3.79525083884723156320009140956, −1.38451886126928944133462712852, 3.60377068539049339525285314324, 4.44164479019975164136856998693, 5.99805830909670243724244795208, 7.67645826150369563852884299287, 9.250410626570516648167872405354, 9.859221823955482145131577321417, 11.23282088130062380188988318623, 11.70306156160910528494019548333, 13.37641206156314875400024776204, 14.93845207445335064089232193024

Graph of the ZZ-function along the critical line