Properties

Label 2-192-16.13-c1-0-1
Degree $2$
Conductor $192$
Sign $0.589 - 0.807i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (−0.334 + 0.334i)5-s + 4.55i·7-s + 1.00i·9-s + (2.47 − 2.47i)11-s + (−0.0594 − 0.0594i)13-s − 0.473·15-s + 3.61·17-s + (−2.55 − 2.55i)19-s + (−3.22 + 3.22i)21-s − 2.82i·23-s + 4.77i·25-s + (−0.707 + 0.707i)27-s + (−5.16 − 5.16i)29-s + 0.557·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.149 + 0.149i)5-s + 1.72i·7-s + 0.333i·9-s + (0.745 − 0.745i)11-s + (−0.0164 − 0.0164i)13-s − 0.122·15-s + 0.877·17-s + (−0.586 − 0.586i)19-s + (−0.703 + 0.703i)21-s − 0.589i·23-s + 0.955i·25-s + (−0.136 + 0.136i)27-s + (−0.958 − 0.958i)29-s + 0.100·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.589 - 0.807i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :1/2),\ 0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17409 + 0.596936i\)
\(L(\frac12)\) \(\approx\) \(1.17409 + 0.596936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.334 - 0.334i)T - 5iT^{2} \)
7 \( 1 - 4.55iT - 7T^{2} \)
11 \( 1 + (-2.47 + 2.47i)T - 11iT^{2} \)
13 \( 1 + (0.0594 + 0.0594i)T + 13iT^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + (2.55 + 2.55i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (5.16 + 5.16i)T + 29iT^{2} \)
31 \( 1 - 0.557T + 31T^{2} \)
37 \( 1 + (-4.38 + 4.38i)T - 37iT^{2} \)
41 \( 1 + 9.27iT - 41T^{2} \)
43 \( 1 + (-1.61 + 1.61i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (0.493 - 0.493i)T - 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (-2.72 - 2.72i)T + 61iT^{2} \)
67 \( 1 + (3.77 + 3.77i)T + 67iT^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 + 0.541iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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

−12.55327591012681365086364136403, −11.72247067700653786803777008253, −10.82170796358521215287844549025, −9.368552449865590050342718848364, −8.912363983307604838953398953482, −7.82576401838318948361709634206, −6.24694480605942912739270865837, −5.30690829790213479006136316781, −3.69853591472143598701898606224, −2.39732776623493553556057084242, 1.38432255337464431038607124454, 3.53948424431201615160273826919, 4.53959882077880813622089041202, 6.39483443843158240436950094343, 7.35939443846833580462171004812, 8.100920191570925532660881592478, 9.538521801529334999233302943317, 10.30337195795278484303486004226, 11.46897971494646271760803602847, 12.54719163938478274563375368570

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