↑

Properties

Label 2-304-19.18-c6-0-51
Degree $2$
Conductor $304$
Sign $-0.999 + 0.0129i$
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  − 36.8i·3-s + 9.31·5-s − 333.·7-s − 629.·9-s + 2.07e3·11-s − 3.60e3i·13-s − 343. i·15-s + 990.·17-s + (6.85e3 − 88.5i)19-s + 1.22e4i·21-s + 9.88e3·23-s − 1.55e4·25-s − 3.65e3i·27-s − 2.96e4i·29-s − 8.08e3i·31-s + ⋯
L(s)  = 1  − 1.36i·3-s + 0.0745·5-s − 0.972·7-s − 0.863·9-s + 1.56·11-s − 1.64i·13-s − 0.101i·15-s + 0.201·17-s + (0.999 − 0.0129i)19-s + 1.32i·21-s + 0.812·23-s − 0.994·25-s − 0.185i·27-s − 1.21i·29-s − 0.271i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0129i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.0129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.999 + 0.0129i$
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: \((2,\ 304,\ (\ :3),\ -0.999 + 0.0129i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.707211266\)
\(L(\frac12)\) \(\approx\) \(1.707211266\)
\(L(4)\) 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 \( 1 + (-6.85e3 + 88.5i)T \)
good3 \( 1 + 36.8iT - 729T^{2} \)
5 \( 1 - 9.31T + 1.56e4T^{2} \)
7 \( 1 + 333.T + 1.17e5T^{2} \)
11 \( 1 - 2.07e3T + 1.77e6T^{2} \)
13 \( 1 + 3.60e3iT - 4.82e6T^{2} \)
17 \( 1 - 990.T + 2.41e7T^{2} \)
23 \( 1 - 9.88e3T + 1.48e8T^{2} \)
29 \( 1 + 2.96e4iT - 5.94e8T^{2} \)
31 \( 1 + 8.08e3iT - 8.87e8T^{2} \)
37 \( 1 + 5.59e3iT - 2.56e9T^{2} \)
41 \( 1 + 5.04e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.04e5T + 6.32e9T^{2} \)
47 \( 1 + 1.87e5T + 1.07e10T^{2} \)
53 \( 1 + 5.64e4iT - 2.21e10T^{2} \)
59 \( 1 - 3.75e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.26e5T + 5.15e10T^{2} \)
67 \( 1 + 3.17e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.75e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.28e5T + 1.51e11T^{2} \)
79 \( 1 + 4.18e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.35e5T + 3.26e11T^{2} \)
89 \( 1 - 6.46e5iT - 4.96e11T^{2} \)
97 \( 1 - 9.34e5iT - 8.32e11T^{2} \)
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

−10.07620344244749899521869836411, −9.294174407865284180345073733959, −8.061623330305573709651918348690, −7.25923760822589198283304799227, −6.38044166226647845136483108358, −5.63005987980837587094558063829, −3.77343868036391755404893850694, −2.69184020984291026748930157065, −1.26890207309105134903085123823, −0.45100958092511830651167630304, 1.42950751990513995093215155181, 3.26441108882998061742211496060, 3.95924151361309489154102611987, 4.95090955086610974326165478297, 6.28488810293202320450695679024, 7.06859422262033766697180462956, 8.838503203477032151924867285103, 9.480101651147347790294302190195, 9.843655702707815730356301546001, 11.18539850436303280471819674765

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