Properties

Label 2-304-19.18-c6-0-47
Degree $2$
Conductor $304$
Sign $0.806 + 0.590i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 50.6i·3-s + 207.·5-s − 216.·7-s − 1.83e3·9-s − 1.26e3·11-s − 2.82e3i·13-s + 1.05e4i·15-s + 3.84e3·17-s + (−5.53e3 − 4.05e3i)19-s − 1.09e4i·21-s − 626.·23-s + 2.73e4·25-s − 5.60e4i·27-s − 2.21e4i·29-s − 5.36e3i·31-s + ⋯
L(s)  = 1  + 1.87i·3-s + 1.65·5-s − 0.631·7-s − 2.51·9-s − 0.953·11-s − 1.28i·13-s + 3.11i·15-s + 0.783·17-s + (−0.806 − 0.590i)19-s − 1.18i·21-s − 0.0515·23-s + 1.75·25-s − 2.84i·27-s − 0.908i·29-s − 0.180i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.806 + 0.590i$
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: \((2,\ 304,\ (\ :3),\ 0.806 + 0.590i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.175893391\)
\(L(\frac12)\) \(\approx\) \(1.175893391\)
\(L(4)\) 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 \( 1 + (5.53e3 + 4.05e3i)T \)
good3 \( 1 - 50.6iT - 729T^{2} \)
5 \( 1 - 207.T + 1.56e4T^{2} \)
7 \( 1 + 216.T + 1.17e5T^{2} \)
11 \( 1 + 1.26e3T + 1.77e6T^{2} \)
13 \( 1 + 2.82e3iT - 4.82e6T^{2} \)
17 \( 1 - 3.84e3T + 2.41e7T^{2} \)
23 \( 1 + 626.T + 1.48e8T^{2} \)
29 \( 1 + 2.21e4iT - 5.94e8T^{2} \)
31 \( 1 + 5.36e3iT - 8.87e8T^{2} \)
37 \( 1 + 4.07e4iT - 2.56e9T^{2} \)
41 \( 1 + 2.60e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.34e5T + 6.32e9T^{2} \)
47 \( 1 + 1.56e5T + 1.07e10T^{2} \)
53 \( 1 - 2.43e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.61e5iT - 4.21e10T^{2} \)
61 \( 1 + 4.34e5T + 5.15e10T^{2} \)
67 \( 1 - 2.17e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.25e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.30e5T + 1.51e11T^{2} \)
79 \( 1 + 4.31e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.18e5T + 3.26e11T^{2} \)
89 \( 1 + 1.89e5iT - 4.96e11T^{2} \)
97 \( 1 - 5.89e5iT - 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48173849465068433214372434881, −9.705093498595410109516877342355, −9.233465850223761823013588551434, −8.020260543966848470505978349427, −6.07867598087935275274435961721, −5.58840961963635036637444320031, −4.68039341752489820721306284573, −3.23484462375153947553772037393, −2.47369444316396222702557701029, −0.25899903523434920246191491594, 1.29359621438740290564581720435, 2.02157433341569991658126713275, 2.93950897182722808984990608544, 5.25931233366795358375797065289, 6.22042914323414075234718308390, 6.63694353300755342052254881744, 7.75826223713102299662498902074, 8.812614757331015657326330004657, 9.754339157884350655472399903712, 10.80426748320398197478620881917

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