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Properties

Label 2-304-19.12-c4-0-32
Degree $2$
Conductor $304$
Sign $0.247 + 0.968i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.7 − 6.76i)3-s + (−15.6 − 27.0i)5-s + 61.1·7-s + (51.0 − 88.4i)9-s + 130.·11-s + (252. + 146. i)13-s + (−365. − 211. i)15-s + (−27.4 − 47.6i)17-s + (276. + 232. i)19-s + (716. − 413. i)21-s + (−96.8 + 167. i)23-s + (−174. + 302. i)25-s − 285. i·27-s + (−856. − 494. i)29-s − 1.40e3i·31-s + ⋯
L(s)  = 1  + (1.30 − 0.751i)3-s + (−0.624 − 1.08i)5-s + 1.24·7-s + (0.630 − 1.09i)9-s + 1.07·11-s + (1.49 + 0.864i)13-s + (−1.62 − 0.938i)15-s + (−0.0950 − 0.164i)17-s + (0.765 + 0.643i)19-s + (1.62 − 0.937i)21-s + (−0.183 + 0.317i)23-s + (−0.279 + 0.483i)25-s − 0.391i·27-s + (−1.01 − 0.587i)29-s − 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.247 + 0.968i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.679295255\)
\(L(\frac12)\) \(\approx\) \(3.679295255\)
\(L(3)\) 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 \( 1 + (-276. - 232. i)T \)
good3 \( 1 + (-11.7 + 6.76i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (15.6 + 27.0i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 61.1T + 2.40e3T^{2} \)
11 \( 1 - 130.T + 1.46e4T^{2} \)
13 \( 1 + (-252. - 146. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (27.4 + 47.6i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (96.8 - 167. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (856. + 494. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.40e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.68e3iT - 1.87e6T^{2} \)
41 \( 1 + (1.38e3 - 800. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-428. - 741. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.03e3 + 1.17e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (2.05e3 - 1.18e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.48e3 - 4.31e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-1.33e3 - 772. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (6.85e3 - 3.95e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-430. - 744. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-6.47e3 + 3.74e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 629.T + 4.74e7T^{2} \)
89 \( 1 + (-2.40e3 - 1.38e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.06e4 + 6.16e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.34406272595104102305132902788, −9.382025307126444191336063175263, −8.818865090551967231658814791721, −8.068206604151106306985498581599, −7.47232538618658128156995731471, −6.00234188023949267968941544439, −4.41914230662378213739924179657, −3.66804092299104678487191327612, −1.78898375644309342444723041753, −1.18325149247128533715365356035, 1.55411703711656482123267838277, 3.20532486227305353854551207866, 3.66068128573965650261016044158, 4.93608587856777804558801373850, 6.58729275741119600550663917877, 7.72633899070500543447201347244, 8.452363862174419949928058728005, 9.179739400542317825149547821261, 10.50708594527383060949863869069, 11.02890299741485381717328044972

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