↑

Properties

Label 2-304-19.18-c6-0-38
Degree $2$
Conductor $304$
Sign $0.881 - 0.472i$
Analytic cond. $69.9364$
Root an. cond. $8.36280$
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  + 27.2i·3-s + 162.·5-s + 68.0·7-s − 11.4·9-s − 838.·11-s − 2.03e3i·13-s + 4.43e3i·15-s + 1.26e3·17-s + (6.04e3 − 3.23e3i)19-s + 1.85e3i·21-s + 2.23e4·23-s + 1.09e4·25-s + 1.95e4i·27-s − 2.15e3i·29-s − 4.25e4i·31-s + ⋯
L(s)  = 1  + 1.00i·3-s + 1.30·5-s + 0.198·7-s − 0.0157·9-s − 0.629·11-s − 0.925i·13-s + 1.31i·15-s + 0.258·17-s + (0.881 − 0.472i)19-s + 0.200i·21-s + 1.83·23-s + 0.700·25-s + 0.991i·27-s − 0.0882i·29-s − 1.42i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(69.9364\)
Root analytic conductor: \(8.36280\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :3),\ 0.881 - 0.472i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.148978054\)
\(L(\frac12)\) \(\approx\) \(3.148978054\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-6.04e3 + 3.23e3i)T \)
good3 \( 1 - 27.2iT - 729T^{2} \)
5 \( 1 - 162.T + 1.56e4T^{2} \)
7 \( 1 - 68.0T + 1.17e5T^{2} \)
11 \( 1 + 838.T + 1.77e6T^{2} \)
13 \( 1 + 2.03e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.26e3T + 2.41e7T^{2} \)
23 \( 1 - 2.23e4T + 1.48e8T^{2} \)
29 \( 1 + 2.15e3iT - 5.94e8T^{2} \)
31 \( 1 + 4.25e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.60e4iT - 2.56e9T^{2} \)
41 \( 1 + 7.83e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.94e4T + 6.32e9T^{2} \)
47 \( 1 - 3.75e4T + 1.07e10T^{2} \)
53 \( 1 + 1.66e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.81e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.73e5T + 5.15e10T^{2} \)
67 \( 1 + 2.73e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.38e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.48e5T + 1.51e11T^{2} \)
79 \( 1 - 5.74e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.44e5T + 3.26e11T^{2} \)
89 \( 1 - 1.32e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.42e5iT - 8.32e11T^{2} \)
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

−10.54709802943748080523640977018, −9.803803382764941546545405203889, −9.274587309279935371478400929579, −8.007515061961926781617547653412, −6.77639822696955579384944019855, −5.36526093829498182415609843236, −5.08632856070649469846052472712, −3.49700839276220706740438221503, −2.38074784094004402708385434531, −0.895930188543114231992059345199, 1.08968465232664344267042636699, 1.81073398060887561510078323602, 2.95934255514513608509976638455, 4.81354532647915380998914443941, 5.78882631426716477952104993301, 6.78564876033241516319250222812, 7.50527587952168184558773874298, 8.757272428980154918740256549964, 9.658608140294896992178140831083, 10.54392901734366273205260854231

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