Properties

Label 2-192-1.1-c7-0-11
Degree $2$
Conductor $192$
Sign $1$
Analytic cond. $59.9779$
Root an. cond. $7.74454$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s + 366.·5-s − 355.·7-s + 729·9-s + 7.10e3·11-s + 3.59e3·13-s + 9.90e3·15-s − 2.13e4·17-s − 1.51e4·19-s − 9.58e3·21-s + 9.53e4·23-s + 5.63e4·25-s + 1.96e4·27-s + 1.94e5·29-s + 1.65e4·31-s + 1.91e5·33-s − 1.30e5·35-s − 3.21e5·37-s + 9.71e4·39-s + 1.59e5·41-s − 8.56e5·43-s + 2.67e5·45-s + 7.48e5·47-s − 6.97e5·49-s − 5.75e5·51-s + 3.05e5·53-s + 2.60e6·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.31·5-s − 0.391·7-s + 0.333·9-s + 1.60·11-s + 0.454·13-s + 0.757·15-s − 1.05·17-s − 0.506·19-s − 0.225·21-s + 1.63·23-s + 0.721·25-s + 0.192·27-s + 1.47·29-s + 0.0995·31-s + 0.928·33-s − 0.513·35-s − 1.04·37-s + 0.262·39-s + 0.360·41-s − 1.64·43-s + 0.437·45-s + 1.05·47-s − 0.846·49-s − 0.607·51-s + 0.281·53-s + 2.11·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $1$
Analytic conductor: \(59.9779\)
Root analytic conductor: \(7.74454\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(3.798329881\)
\(L(\frac12)\) \(\approx\) \(3.798329881\)
\(L(\frac{9}{2})\) 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 \( 1 - 27T \)
good5 \( 1 - 366.T + 7.81e4T^{2} \)
7 \( 1 + 355.T + 8.23e5T^{2} \)
11 \( 1 - 7.10e3T + 1.94e7T^{2} \)
13 \( 1 - 3.59e3T + 6.27e7T^{2} \)
17 \( 1 + 2.13e4T + 4.10e8T^{2} \)
19 \( 1 + 1.51e4T + 8.93e8T^{2} \)
23 \( 1 - 9.53e4T + 3.40e9T^{2} \)
29 \( 1 - 1.94e5T + 1.72e10T^{2} \)
31 \( 1 - 1.65e4T + 2.75e10T^{2} \)
37 \( 1 + 3.21e5T + 9.49e10T^{2} \)
41 \( 1 - 1.59e5T + 1.94e11T^{2} \)
43 \( 1 + 8.56e5T + 2.71e11T^{2} \)
47 \( 1 - 7.48e5T + 5.06e11T^{2} \)
53 \( 1 - 3.05e5T + 1.17e12T^{2} \)
59 \( 1 + 6.88e5T + 2.48e12T^{2} \)
61 \( 1 + 1.17e6T + 3.14e12T^{2} \)
67 \( 1 - 4.25e6T + 6.06e12T^{2} \)
71 \( 1 - 4.43e6T + 9.09e12T^{2} \)
73 \( 1 + 2.12e6T + 1.10e13T^{2} \)
79 \( 1 - 2.37e6T + 1.92e13T^{2} \)
83 \( 1 - 6.71e6T + 2.71e13T^{2} \)
89 \( 1 + 5.65e6T + 4.42e13T^{2} \)
97 \( 1 - 1.56e7T + 8.07e13T^{2} \)
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

−11.12582050872074634351333637456, −10.06265544433597835095918259566, −9.153226138170066288431897094107, −8.650488479886691090018619496246, −6.78233690639230427281150553866, −6.34916094324655108058132357688, −4.81301263676893810713655399505, −3.46467195282207751725602114736, −2.17541138333839925490682948960, −1.11548648159647414790566685043, 1.11548648159647414790566685043, 2.17541138333839925490682948960, 3.46467195282207751725602114736, 4.81301263676893810713655399505, 6.34916094324655108058132357688, 6.78233690639230427281150553866, 8.650488479886691090018619496246, 9.153226138170066288431897094107, 10.06265544433597835095918259566, 11.12582050872074634351333637456

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