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Properties

Label 2-304-19.18-c4-0-8
Degree $2$
Conductor $304$
Sign $0.969 - 0.246i$
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  − 11.4i·3-s − 9.10·5-s − 75.9·7-s − 49.6·9-s − 149.·11-s + 188. i·13-s + 104. i·15-s + 322.·17-s + (349. − 89.0i)19-s + 867. i·21-s + 561.·23-s − 542.·25-s − 358. i·27-s + 260. i·29-s + 167. i·31-s + ⋯
L(s)  = 1  − 1.27i·3-s − 0.364·5-s − 1.54·7-s − 0.612·9-s − 1.23·11-s + 1.11i·13-s + 0.462i·15-s + 1.11·17-s + (0.969 − 0.246i)19-s + 1.96i·21-s + 1.06·23-s − 0.867·25-s − 0.491i·27-s + 0.310i·29-s + 0.174i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.969 - 0.246i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.969 - 0.246i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9357886735\)
\(L(\frac12)\) \(\approx\) \(0.9357886735\)
\(L(3)\) 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 \( 1 + (-349. + 89.0i)T \)
good3 \( 1 + 11.4iT - 81T^{2} \)
5 \( 1 + 9.10T + 625T^{2} \)
7 \( 1 + 75.9T + 2.40e3T^{2} \)
11 \( 1 + 149.T + 1.46e4T^{2} \)
13 \( 1 - 188. iT - 2.85e4T^{2} \)
17 \( 1 - 322.T + 8.35e4T^{2} \)
23 \( 1 - 561.T + 2.79e5T^{2} \)
29 \( 1 - 260. iT - 7.07e5T^{2} \)
31 \( 1 - 167. iT - 9.23e5T^{2} \)
37 \( 1 + 1.80e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.42e3iT - 2.82e6T^{2} \)
43 \( 1 + 836.T + 3.41e6T^{2} \)
47 \( 1 + 1.08e3T + 4.87e6T^{2} \)
53 \( 1 - 4.33e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.83e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.51e3T + 1.38e7T^{2} \)
67 \( 1 - 3.10e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.92e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.35e3T + 2.83e7T^{2} \)
79 \( 1 - 8.36e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.10e4T + 4.74e7T^{2} \)
89 \( 1 - 834. iT - 6.27e7T^{2} \)
97 \( 1 + 2.25e3iT - 8.85e7T^{2} \)
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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

−11.36195705568999635408301723596, −10.06356259911745310011318804763, −9.274411610623919507895038847813, −7.928398346140195954151604765453, −7.21788822989331951736708727556, −6.46577302054230501100136933304, −5.32820350678161635530408532923, −3.56826082034593241454489227355, −2.48629833489687779417666047999, −0.902287291979255123869243762453, 0.37775298993572420015611913539, 3.07906490348259270646405101400, 3.49573252392588944238686920390, 5.02225689271862662551491741060, 5.78182303050479014941417666381, 7.25924566183823888818052915785, 8.242468603856203285897702472137, 9.604317491661543515757177042744, 9.995796665904167369763028744626, 10.68163553667865182588325934265

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