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Properties

Label 2-192-16.13-c3-0-4
Degree $2$
Conductor $192$
Sign $-0.136 - 0.990i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.12 + 2.12i)3-s + (3.22 − 3.22i)5-s + 24.6i·7-s + 8.99i·9-s + (−23.7 + 23.7i)11-s + (−18.6 − 18.6i)13-s + 13.6·15-s − 3.55·17-s + (109. + 109. i)19-s + (−52.2 + 52.2i)21-s − 36.5i·23-s + 104. i·25-s + (−19.0 + 19.0i)27-s + (68.8 + 68.8i)29-s − 306.·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.288 − 0.288i)5-s + 1.32i·7-s + 0.333i·9-s + (−0.651 + 0.651i)11-s + (−0.398 − 0.398i)13-s + 0.235·15-s − 0.0506·17-s + (1.31 + 1.31i)19-s + (−0.542 + 0.542i)21-s − 0.331i·23-s + 0.833i·25-s + (−0.136 + 0.136i)27-s + (0.440 + 0.440i)29-s − 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.136 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.15121 + 1.32100i\)
\(L(\frac12)\) \(\approx\) \(1.15121 + 1.32100i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.12 - 2.12i)T \)
good5 \( 1 + (-3.22 + 3.22i)T - 125iT^{2} \)
7 \( 1 - 24.6iT - 343T^{2} \)
11 \( 1 + (23.7 - 23.7i)T - 1.33e3iT^{2} \)
13 \( 1 + (18.6 + 18.6i)T + 2.19e3iT^{2} \)
17 \( 1 + 3.55T + 4.91e3T^{2} \)
19 \( 1 + (-109. - 109. i)T + 6.85e3iT^{2} \)
23 \( 1 + 36.5iT - 1.21e4T^{2} \)
29 \( 1 + (-68.8 - 68.8i)T + 2.43e4iT^{2} \)
31 \( 1 + 306.T + 2.97e4T^{2} \)
37 \( 1 + (-92.9 + 92.9i)T - 5.06e4iT^{2} \)
41 \( 1 - 385. iT - 6.89e4T^{2} \)
43 \( 1 + (150. - 150. i)T - 7.95e4iT^{2} \)
47 \( 1 - 114.T + 1.03e5T^{2} \)
53 \( 1 + (-451. + 451. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-544. + 544. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-179. - 179. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-283. - 283. i)T + 3.00e5iT^{2} \)
71 \( 1 + 930. iT - 3.57e5T^{2} \)
73 \( 1 + 701. iT - 3.89e5T^{2} \)
79 \( 1 + 779.T + 4.93e5T^{2} \)
83 \( 1 + (-296. - 296. i)T + 5.71e5iT^{2} \)
89 \( 1 + 865. iT - 7.04e5T^{2} \)
97 \( 1 + 542.T + 9.12e5T^{2} \)
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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

−12.46544767881010919776496628903, −11.44245520954264767524252913411, −10.08714353443336019702520925364, −9.442879298103763600744153238502, −8.435075668946937784016344642396, −7.42578213280957414751693619014, −5.70283252242505469835666091476, −5.03227839052380305240328244777, −3.27549715969744152989787563613, −1.98215776974221411896230085771, 0.73311045421030142295763130830, 2.56980105500589687475835171874, 3.92036369845813433491704617675, 5.42945165888432756688016977814, 6.95150052990113125927522148062, 7.47374563058770393521876231947, 8.770367732832938436455634480491, 9.897939106528267056433158198419, 10.77972011855509677358531048771, 11.75804606743720222881451068860

Graph of the $Z$-function along the critical line