Properties

Label 2-192-8.5-c3-0-5
Degree $2$
Conductor $192$
Sign $0.965 + 0.258i$
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  − 3i·3-s + 3.46i·5-s + 24.2·7-s − 9·9-s + 48i·11-s − 41.5i·13-s + 10.3·15-s + 54·17-s + 4i·19-s − 72.7i·21-s + 173.·23-s + 113·25-s + 27i·27-s − 162. i·29-s + 58.8·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.309i·5-s + 1.30·7-s − 0.333·9-s + 1.31i·11-s − 0.886i·13-s + 0.178·15-s + 0.770·17-s + 0.0482i·19-s − 0.755i·21-s + 1.57·23-s + 0.904·25-s + 0.192i·27-s − 1.04i·29-s + 0.341·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.965 + 0.258i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.00643 - 0.264152i\)
\(L(\frac12)\) \(\approx\) \(2.00643 - 0.264152i\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
good5 \( 1 - 3.46iT - 125T^{2} \)
7 \( 1 - 24.2T + 343T^{2} \)
11 \( 1 - 48iT - 1.33e3T^{2} \)
13 \( 1 + 41.5iT - 2.19e3T^{2} \)
17 \( 1 - 54T + 4.91e3T^{2} \)
19 \( 1 - 4iT - 6.85e3T^{2} \)
23 \( 1 - 173.T + 1.21e4T^{2} \)
29 \( 1 + 162. iT - 2.43e4T^{2} \)
31 \( 1 - 58.8T + 2.97e4T^{2} \)
37 \( 1 + 325. iT - 5.06e4T^{2} \)
41 \( 1 + 294T + 6.89e4T^{2} \)
43 \( 1 - 188iT - 7.95e4T^{2} \)
47 \( 1 - 505.T + 1.03e5T^{2} \)
53 \( 1 - 744. iT - 1.48e5T^{2} \)
59 \( 1 - 252iT - 2.05e5T^{2} \)
61 \( 1 - 90.0iT - 2.26e5T^{2} \)
67 \( 1 + 628iT - 3.00e5T^{2} \)
71 \( 1 + 6.92T + 3.57e5T^{2} \)
73 \( 1 + 1.00e3T + 3.89e5T^{2} \)
79 \( 1 + 1.34e3T + 4.93e5T^{2} \)
83 \( 1 - 720iT - 5.71e5T^{2} \)
89 \( 1 + 1.48e3T + 7.04e5T^{2} \)
97 \( 1 - 1.82e3T + 9.12e5T^{2} \)
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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

−12.12558169297969469295506078846, −11.09520441905877589515366024542, −10.24184191282210251100291666770, −8.888572529015301387636675204298, −7.74016892317729523793644520842, −7.16634662060996807894710402171, −5.62342742433537457305644356755, −4.55380105749821572389403599080, −2.69348918937598456249345726662, −1.24878618868391497694858730949, 1.23184531978919610774676607727, 3.20261935189415211060644030745, 4.66810372483814567510359672167, 5.45267556693726555884314250413, 6.98598970870768570764569196367, 8.447021118912845150489565926778, 8.842873712942778089841656892782, 10.28790349129504435599086789473, 11.22289682789172044079495797682, 11.79074829856699289253686857069

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