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Properties

Label 2-192-16.5-c1-0-1
Degree $2$
Conductor $192$
Sign $0.987 - 0.154i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.987 - 0.154i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.987 - 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45441 + 0.113184i\)
\(L(\frac12)\) \(\approx\) \(1.45441 + 0.113184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-1.74 - 1.74i)T + 5iT^{2} \)
7 \( 1 - 2.55iT - 7T^{2} \)
11 \( 1 + (0.473 + 0.473i)T + 11iT^{2} \)
13 \( 1 + (-2.88 + 2.88i)T - 13iT^{2} \)
17 \( 1 + 6.44T + 17T^{2} \)
19 \( 1 + (-4.55 + 4.55i)T - 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (3.07 - 3.07i)T - 29iT^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 + (2.72 + 2.72i)T + 37iT^{2} \)
41 \( 1 + 0.788iT - 41T^{2} \)
43 \( 1 + (-0.389 - 0.389i)T + 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (2.57 + 2.57i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (4.38 - 4.38i)T - 61iT^{2} \)
67 \( 1 + (-2.11 + 2.11i)T - 67iT^{2} \)
71 \( 1 + 5.11iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 - 6.31T + 79T^{2} \)
83 \( 1 + (0.641 - 0.641i)T - 83iT^{2} \)
89 \( 1 + 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line