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Properties

Label 2-192-16.5-c1-0-1
Degree 22
Conductor 192192
Sign 0.987−0.154i0.987 - 0.154i
Analytic cond. 1.533121.53312
Root an. cond. 1.238191.23819
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  + (0.707 − 0.707i)3-s + (1.74 + 1.74i)5-s + 2.55i·7-s − 1.00i·9-s + (−0.473 − 0.473i)11-s + (2.88 − 2.88i)13-s + 2.47·15-s − 6.44·17-s + (4.55 − 4.55i)19-s + (1.80 + 1.80i)21-s + 2.82i·23-s + 1.11i·25-s + (−0.707 − 0.707i)27-s + (−3.07 + 3.07i)29-s − 6.55·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.782 + 0.782i)5-s + 0.966i·7-s − 0.333i·9-s + (−0.142 − 0.142i)11-s + (0.800 − 0.800i)13-s + 0.638·15-s − 1.56·17-s + (1.04 − 1.04i)19-s + (0.394 + 0.394i)21-s + 0.589i·23-s + 0.223i·25-s + (−0.136 − 0.136i)27-s + (−0.571 + 0.571i)29-s − 1.17·31-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=((0.987−0.154i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+1/2) L(s)=((0.987−0.154i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: 0.987−0.154i0.987 - 0.154i
Analytic conductor: 1.533121.53312
Root analytic conductor: 1.238191.23819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ192(49,⋅)\chi_{192} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 192, ( :1/2), 0.987−0.154i)(2,\ 192,\ (\ :1/2),\ 0.987 - 0.154i)

Particular Values

L(1)L(1) ≈\approx 1.45441+0.113184i1.45441 + 0.113184i
L(12)L(\frac12) ≈\approx 1.45441+0.113184i1.45441 + 0.113184i
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
3 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good5 1+(−1.74−1.74i)T+5iT2 1 + (-1.74 - 1.74i)T + 5iT^{2}
7 1−2.55iT−7T2 1 - 2.55iT - 7T^{2}
11 1+(0.473+0.473i)T+11iT2 1 + (0.473 + 0.473i)T + 11iT^{2}
13 1+(−2.88+2.88i)T−13iT2 1 + (-2.88 + 2.88i)T - 13iT^{2}
17 1+6.44T+17T2 1 + 6.44T + 17T^{2}
19 1+(−4.55+4.55i)T−19iT2 1 + (-4.55 + 4.55i)T - 19iT^{2}
23 1−2.82iT−23T2 1 - 2.82iT - 23T^{2}
29 1+(3.07−3.07i)T−29iT2 1 + (3.07 - 3.07i)T - 29iT^{2}
31 1+6.55T+31T2 1 + 6.55T + 31T^{2}
37 1+(2.72+2.72i)T+37iT2 1 + (2.72 + 2.72i)T + 37iT^{2}
41 1+0.788iT−41T2 1 + 0.788iT - 41T^{2}
43 1+(−0.389−0.389i)T+43iT2 1 + (-0.389 - 0.389i)T + 43iT^{2}
47 1+2.82T+47T2 1 + 2.82T + 47T^{2}
53 1+(2.57+2.57i)T+53iT2 1 + (2.57 + 2.57i)T + 53iT^{2}
59 1+(4+4i)T+59iT2 1 + (4 + 4i)T + 59iT^{2}
61 1+(4.38−4.38i)T−61iT2 1 + (4.38 - 4.38i)T - 61iT^{2}
67 1+(−2.11+2.11i)T−67iT2 1 + (-2.11 + 2.11i)T - 67iT^{2}
71 1+5.11iT−71T2 1 + 5.11iT - 71T^{2}
73 1−14.7iT−73T2 1 - 14.7iT - 73T^{2}
79 1−6.31T+79T2 1 - 6.31T + 79T^{2}
83 1+(0.641−0.641i)T−83iT2 1 + (0.641 - 0.641i)T - 83iT^{2}
89 1+6.31iT−89T2 1 + 6.31iT - 89T^{2}
97 1−12.6T+97T2 1 - 12.6T + 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

−12.83410427350285364867725261906, −11.46951032849543076039419169348, −10.71172509691700081387673533343, −9.379750123936178301407461322060, −8.720042663345123197762826967454, −7.35860026373326499674118121572, −6.31592504371531756545917644584, −5.34553772472817675156644250434, −3.25287641008141629079339590286, −2.14354634345310030434999238698, 1.75006188128523355498547120739, 3.77672130440115658931169717981, 4.82758846500107618268426864068, 6.17585035869479604888751508001, 7.47610642499185956298168912329, 8.750384106499168649395141159630, 9.441091881585600621519985659515, 10.43759532135277580242342675059, 11.38393810595461787892514443741, 12.78932271067515571647797913886

Graph of the ZZ-function along the critical line