↑

Properties

Label 2-304-19.11-c5-0-5
Degree $2$
Conductor $304$
Sign $-0.712 + 0.701i$
Analytic cond. $48.7566$
Root an. cond. $6.98259$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−12.4 + 21.5i)3-s + (−18.0 + 31.3i)5-s + 208.·7-s + (−187. − 324. i)9-s + 204.·11-s + (−146. − 254. i)13-s + (−449. − 779. i)15-s + (−850. + 1.47e3i)17-s + (−1.32e3 + 841. i)19-s + (−2.59e3 + 4.49e3i)21-s + (1.76e3 + 3.06e3i)23-s + (907. + 1.57e3i)25-s + 3.28e3·27-s + (−3.20e3 − 5.55e3i)29-s + 2.73e3·31-s + ⋯
L(s)  = 1  + (−0.797 + 1.38i)3-s + (−0.323 + 0.560i)5-s + 1.61·7-s + (−0.771 − 1.33i)9-s + 0.509·11-s + (−0.240 − 0.417i)13-s + (−0.516 − 0.894i)15-s + (−0.713 + 1.23i)17-s + (−0.844 + 0.535i)19-s + (−1.28 + 2.22i)21-s + (0.696 + 1.20i)23-s + (0.290 + 0.503i)25-s + 0.867·27-s + (−0.707 − 1.22i)29-s + 0.510·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.712 + 0.701i$
Analytic conductor: \(48.7566\)
Root analytic conductor: \(6.98259\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :5/2),\ -0.712 + 0.701i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8258496773\)
\(L(\frac12)\) \(\approx\) \(0.8258496773\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (1.32e3 - 841. i)T \)
good3 \( 1 + (12.4 - 21.5i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (18.0 - 31.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 - 208.T + 1.68e4T^{2} \)
11 \( 1 - 204.T + 1.61e5T^{2} \)
13 \( 1 + (146. + 254. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (850. - 1.47e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (-1.76e3 - 3.06e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (3.20e3 + 5.55e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 - 2.73e3T + 2.86e7T^{2} \)
37 \( 1 + 5.42e3T + 6.93e7T^{2} \)
41 \( 1 + (3.69e3 - 6.39e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (7.87e3 - 1.36e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (9.72e3 + 1.68e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.29e3 + 7.43e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.39e4 - 2.41e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.53e4 + 2.66e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.41e4 - 4.17e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.43e3 + 5.94e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-1.93e4 + 3.34e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.21e4 + 5.56e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 7.15e4T + 3.93e9T^{2} \)
89 \( 1 + (2.32e4 + 4.02e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (4.52e4 - 7.82e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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

−11.25969694455143254319936725485, −10.73386713068523861900559226315, −9.863666586834587792012170157498, −8.685346997050333471049570420017, −7.75811934958237506308424286838, −6.35067883816917002990480577496, −5.27964453503448765498198281158, −4.44462784559550889963454244796, −3.57622435094669537821582452614, −1.67117131229118681416416218170, 0.25824780312879803766288191070, 1.27505483320974880471894553627, 2.25504857309452807383032221600, 4.57169959648958641142203895814, 5.12812162122040948520008577447, 6.61404216835313186601908618383, 7.20050583209833549375694855883, 8.306018163104306662052143694248, 8.953409722783035705523713381672, 10.84733935920664755263715715873

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