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Properties

Label 2-304-19.18-c6-0-4
Degree $2$
Conductor $304$
Sign $-0.646 + 0.763i$
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  + 21.6i·3-s − 216.·5-s + 134.·7-s + 260.·9-s + 610.·11-s + 3.17e3i·13-s − 4.69e3i·15-s − 4.96e3·17-s + (−4.43e3 + 5.23e3i)19-s + 2.90e3i·21-s + 1.09e4·23-s + 3.13e4·25-s + 2.14e4i·27-s + 4.36e4i·29-s − 2.14e3i·31-s + ⋯
L(s)  = 1  + 0.801i·3-s − 1.73·5-s + 0.391·7-s + 0.357·9-s + 0.458·11-s + 1.44i·13-s − 1.39i·15-s − 1.00·17-s + (−0.646 + 0.763i)19-s + 0.313i·21-s + 0.901·23-s + 2.00·25-s + 1.08i·27-s + 1.79i·29-s − 0.0720i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.646 + 0.763i$
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: \((2,\ 304,\ (\ :3),\ -0.646 + 0.763i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.4425714324\)
\(L(\frac12)\) \(\approx\) \(0.4425714324\)
\(L(4)\) 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 \( 1 + (4.43e3 - 5.23e3i)T \)
good3 \( 1 - 21.6iT - 729T^{2} \)
5 \( 1 + 216.T + 1.56e4T^{2} \)
7 \( 1 - 134.T + 1.17e5T^{2} \)
11 \( 1 - 610.T + 1.77e6T^{2} \)
13 \( 1 - 3.17e3iT - 4.82e6T^{2} \)
17 \( 1 + 4.96e3T + 2.41e7T^{2} \)
23 \( 1 - 1.09e4T + 1.48e8T^{2} \)
29 \( 1 - 4.36e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.14e3iT - 8.87e8T^{2} \)
37 \( 1 + 3.22e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.44e3iT - 4.75e9T^{2} \)
43 \( 1 + 6.40e4T + 6.32e9T^{2} \)
47 \( 1 - 9.69e4T + 1.07e10T^{2} \)
53 \( 1 - 5.85e4iT - 2.21e10T^{2} \)
59 \( 1 - 7.88e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.24e4T + 5.15e10T^{2} \)
67 \( 1 + 5.57e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.75e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.78e5T + 1.51e11T^{2} \)
79 \( 1 - 4.41e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.86e5T + 3.26e11T^{2} \)
89 \( 1 + 3.72e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.92e5iT - 8.32e11T^{2} \)
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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

−11.17433462516179709237081403559, −10.64435772862831832175306620919, −9.180850969736802392296260268382, −8.643133193905560675446808555168, −7.40605664106642544360368685530, −6.69028405020266956874206347472, −4.78665106665582942246187467252, −4.25659320969576067230051188994, −3.44751607058421281624341797611, −1.56024340187892729015646386807, 0.13130356088235654440610620964, 0.998034160558047931286813320855, 2.62429523262614555614377049616, 3.94727897093049071026249994090, 4.80822285557731763876390687219, 6.46367370626543878070536166587, 7.32252395013342751392871554910, 8.020458038879991334727936221002, 8.747173198857441635738505109061, 10.30335144189159507728776652483

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