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Properties

Label 2-92-23.2-c1-0-0
Degree 22
Conductor 9292
Sign 0.729−0.684i0.729 - 0.684i
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  + (0.210 + 1.46i)3-s + (0.926 + 0.272i)5-s + (−1.14 + 1.32i)7-s + (0.785 − 0.230i)9-s + (0.436 − 0.280i)11-s + (−2.04 − 2.35i)13-s + (−0.203 + 1.41i)15-s + (1.39 − 3.05i)17-s + (−2.90 − 6.36i)19-s + (−2.17 − 1.40i)21-s + (−1.35 + 4.60i)23-s + (−3.42 − 2.19i)25-s + (2.34 + 5.13i)27-s + (1.01 − 2.21i)29-s + (−0.253 + 1.76i)31-s + ⋯
L(s)  = 1  + (0.121 + 0.843i)3-s + (0.414 + 0.121i)5-s + (−0.434 + 0.501i)7-s + (0.261 − 0.0768i)9-s + (0.131 − 0.0845i)11-s + (−0.566 − 0.654i)13-s + (−0.0524 + 0.364i)15-s + (0.338 − 0.740i)17-s + (−0.666 − 1.46i)19-s + (−0.475 − 0.305i)21-s + (−0.282 + 0.959i)23-s + (−0.684 − 0.439i)25-s + (0.450 + 0.987i)27-s + (0.188 − 0.412i)29-s + (−0.0454 + 0.316i)31-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((0.729−0.684i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+1/2) L(s)=((0.729−0.684i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: 0.729−0.684i0.729 - 0.684i
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), 0.729−0.684i)(2,\ 92,\ (\ :1/2),\ 0.729 - 0.684i)

Particular Values

L(1)L(1) ≈\approx 0.974050+0.385623i0.974050 + 0.385623i
L(12)L(\frac12) ≈\approx 0.974050+0.385623i0.974050 + 0.385623i
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+(1.35−4.60i)T 1 + (1.35 - 4.60i)T
good3 1+(−0.210−1.46i)T+(−2.87+0.845i)T2 1 + (-0.210 - 1.46i)T + (-2.87 + 0.845i)T^{2}
5 1+(−0.926−0.272i)T+(4.20+2.70i)T2 1 + (-0.926 - 0.272i)T + (4.20 + 2.70i)T^{2}
7 1+(1.14−1.32i)T+(−0.996−6.92i)T2 1 + (1.14 - 1.32i)T + (-0.996 - 6.92i)T^{2}
11 1+(−0.436+0.280i)T+(4.56−10.0i)T2 1 + (-0.436 + 0.280i)T + (4.56 - 10.0i)T^{2}
13 1+(2.04+2.35i)T+(−1.85+12.8i)T2 1 + (2.04 + 2.35i)T + (-1.85 + 12.8i)T^{2}
17 1+(−1.39+3.05i)T+(−11.1−12.8i)T2 1 + (-1.39 + 3.05i)T + (-11.1 - 12.8i)T^{2}
19 1+(2.90+6.36i)T+(−12.4+14.3i)T2 1 + (2.90 + 6.36i)T + (-12.4 + 14.3i)T^{2}
29 1+(−1.01+2.21i)T+(−18.9−21.9i)T2 1 + (-1.01 + 2.21i)T + (-18.9 - 21.9i)T^{2}
31 1+(0.253−1.76i)T+(−29.7−8.73i)T2 1 + (0.253 - 1.76i)T + (-29.7 - 8.73i)T^{2}
37 1+(4.58−1.34i)T+(31.1−20.0i)T2 1 + (4.58 - 1.34i)T + (31.1 - 20.0i)T^{2}
41 1+(−12.0−3.54i)T+(34.4+22.1i)T2 1 + (-12.0 - 3.54i)T + (34.4 + 22.1i)T^{2}
43 1+(0.236+1.64i)T+(−41.2+12.1i)T2 1 + (0.236 + 1.64i)T + (-41.2 + 12.1i)T^{2}
47 1−5.46T+47T2 1 - 5.46T + 47T^{2}
53 1+(8.08−9.32i)T+(−7.54−52.4i)T2 1 + (8.08 - 9.32i)T + (-7.54 - 52.4i)T^{2}
59 1+(0.434+0.500i)T+(−8.39+58.3i)T2 1 + (0.434 + 0.500i)T + (-8.39 + 58.3i)T^{2}
61 1+(0.685−4.76i)T+(−58.5−17.1i)T2 1 + (0.685 - 4.76i)T + (-58.5 - 17.1i)T^{2}
67 1+(9.51+6.11i)T+(27.8+60.9i)T2 1 + (9.51 + 6.11i)T + (27.8 + 60.9i)T^{2}
71 1+(4.30+2.76i)T+(29.4+64.5i)T2 1 + (4.30 + 2.76i)T + (29.4 + 64.5i)T^{2}
73 1+(−5.40−11.8i)T+(−47.8+55.1i)T2 1 + (-5.40 - 11.8i)T + (-47.8 + 55.1i)T^{2}
79 1+(9.56+11.0i)T+(−11.2+78.1i)T2 1 + (9.56 + 11.0i)T + (-11.2 + 78.1i)T^{2}
83 1+(−2.91+0.854i)T+(69.8−44.8i)T2 1 + (-2.91 + 0.854i)T + (69.8 - 44.8i)T^{2}
89 1+(1.78+12.4i)T+(−85.3+25.0i)T2 1 + (1.78 + 12.4i)T + (-85.3 + 25.0i)T^{2}
97 1+(0.645+0.189i)T+(81.6+52.4i)T2 1 + (0.645 + 0.189i)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

−14.29643599728722020389167437357, −13.17792279286181374605871329143, −12.09424949287418530683732839216, −10.76083190376504744449616307848, −9.738496449049352445334333549252, −9.073880864857998544431910803267, −7.38914735333399448557058400501, −5.90621966047291652155226060729, −4.55649725117800984101633133659, −2.87271157771041116109145278792, 1.88651142605817688110252411733, 4.10055755733027104876728142176, 6.01656916805630417788144171389, 7.07764940851475640223032654103, 8.179486869148536406465343017569, 9.655645674845707814589151454928, 10.57663904045988798365334061599, 12.23733338297074728087565847737, 12.79064336001435995505304768343, 13.87746279969656257270825461503

Graph of the ZZ-function along the critical line