↑

Properties

Label 2-192-16.11-c6-0-13
Degree $2$
Conductor $192$
Sign $0.720 - 0.693i$
Analytic cond. $44.1703$
Root an. cond. $6.64608$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (11.0 + 11.0i)3-s + (52.2 + 52.2i)5-s + 282.·7-s + 242. i·9-s + (135. − 135. i)11-s + (730. − 730. i)13-s + 1.15e3i·15-s + 8.44e3·17-s + (1.83e3 + 1.83e3i)19-s + (3.11e3 + 3.11e3i)21-s − 1.44e4·23-s − 1.01e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (2.25e4 − 2.25e4i)29-s + 2.57e4i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.417 + 0.417i)5-s + 0.823·7-s + 0.333i·9-s + (0.102 − 0.102i)11-s + (0.332 − 0.332i)13-s + 0.341i·15-s + 1.71·17-s + (0.267 + 0.267i)19-s + (0.336 + 0.336i)21-s − 1.18·23-s − 0.651i·25-s + (−0.136 + 0.136i)27-s + (0.923 − 0.923i)29-s + 0.864i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(44.1703\)
Root analytic conductor: \(6.64608\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.116995808\)
\(L(\frac12)\) \(\approx\) \(3.116995808\)
\(L(4)\) 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 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (-52.2 - 52.2i)T + 1.56e4iT^{2} \)
7 \( 1 - 282.T + 1.17e5T^{2} \)
11 \( 1 + (-135. + 135. i)T - 1.77e6iT^{2} \)
13 \( 1 + (-730. + 730. i)T - 4.82e6iT^{2} \)
17 \( 1 - 8.44e3T + 2.41e7T^{2} \)
19 \( 1 + (-1.83e3 - 1.83e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.44e4T + 1.48e8T^{2} \)
29 \( 1 + (-2.25e4 + 2.25e4i)T - 5.94e8iT^{2} \)
31 \( 1 - 2.57e4iT - 8.87e8T^{2} \)
37 \( 1 + (-2.26e4 - 2.26e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 1.85e4iT - 4.75e9T^{2} \)
43 \( 1 + (1.82e4 - 1.82e4i)T - 6.32e9iT^{2} \)
47 \( 1 + 4.38e4iT - 1.07e10T^{2} \)
53 \( 1 + (-1.50e5 - 1.50e5i)T + 2.21e10iT^{2} \)
59 \( 1 + (1.69e5 - 1.69e5i)T - 4.21e10iT^{2} \)
61 \( 1 + (-2.12e5 + 2.12e5i)T - 5.15e10iT^{2} \)
67 \( 1 + (6.01e4 + 6.01e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 4.74e5T + 1.28e11T^{2} \)
73 \( 1 - 6.41e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.15e5iT - 2.43e11T^{2} \)
83 \( 1 + (-1.79e5 - 1.79e5i)T + 3.26e11iT^{2} \)
89 \( 1 - 5.93e5iT - 4.96e11T^{2} \)
97 \( 1 - 7.72e5T + 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.52676296183624491228255181436, −10.32975315956425330040169691278, −9.844292377836611585926274058752, −8.390703951908596534796451327573, −7.77884468974284733162115142397, −6.24862763699990467182095196177, −5.18552735315273711571468827680, −3.85615674552645322742643177133, −2.61635092164406974127210560937, −1.20225699698433688629856118113, 0.996516330847213964612109530238, 1.97589943036464897755888829408, 3.54438159899677075622774178196, 4.97380571496598311178881976224, 6.03331015507076522268821430281, 7.43126582084779352192162087882, 8.234119767449288039846724069233, 9.254545647974853652438475985207, 10.21873459521341419004677602902, 11.51849786931254320399846664925

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