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Properties

Label 2-304-19.12-c4-0-24
Degree $2$
Conductor $304$
Sign $0.946 + 0.323i$
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  + (−10.6 + 6.12i)3-s + (3.27 + 5.66i)5-s + 87.4·7-s + (34.5 − 59.8i)9-s + 159.·11-s + (−223. − 128. i)13-s + (−69.4 − 40.0i)15-s + (−151. − 263. i)17-s + (−345. + 103. i)19-s + (−928. + 536. i)21-s + (−60.7 + 105. i)23-s + (291. − 504. i)25-s − 145. i·27-s + (1.03e3 + 596. i)29-s − 1.12e3i·31-s + ⋯
L(s)  = 1  + (−1.17 + 0.680i)3-s + (0.130 + 0.226i)5-s + 1.78·7-s + (0.426 − 0.739i)9-s + 1.32·11-s + (−1.32 − 0.763i)13-s + (−0.308 − 0.178i)15-s + (−0.525 − 0.910i)17-s + (−0.957 + 0.287i)19-s + (−2.10 + 1.21i)21-s + (−0.114 + 0.198i)23-s + (0.465 − 0.806i)25-s − 0.199i·27-s + (1.22 + 0.708i)29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.946 + 0.323i$
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: \((2,\ 304,\ (\ :2),\ 0.946 + 0.323i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.444876958\)
\(L(\frac12)\) \(\approx\) \(1.444876958\)
\(L(3)\) 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 \( 1 + (345. - 103. i)T \)
good3 \( 1 + (10.6 - 6.12i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-3.27 - 5.66i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 87.4T + 2.40e3T^{2} \)
11 \( 1 - 159.T + 1.46e4T^{2} \)
13 \( 1 + (223. + 128. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (151. + 263. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (60.7 - 105. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.03e3 - 596. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 1.12e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.46e3iT - 1.87e6T^{2} \)
41 \( 1 + (700. - 404. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.06e3 - 1.84e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.57e3 + 2.72e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.10e3 + 637. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (2.53e3 - 1.46e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.30e3 - 2.26e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-642. - 370. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-5.88e3 + 3.39e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-422. - 732. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-4.66e3 + 2.69e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 2.13e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.21e3 - 700. i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-749. + 432. i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−11.01088548138898758921632995624, −10.41325421913734874170278071307, −9.307555238495258671305434196245, −8.153311890084630733637361614002, −7.02876235892297910487130592885, −5.86227979018115216898340245052, −4.85027875584578607894126296578, −4.32336856662750757330902563761, −2.23353687105673806162996194981, −0.61290457493951553933076230354, 1.12231645887619556422994360947, 1.95088461666501645293885638614, 4.42335327975125796520260900273, 5.00142943073078688020166842264, 6.33596124478263871628988344219, 6.99558887237124873064741959265, 8.214553568481995376421237330587, 9.111783226298899520925517370269, 10.58023388058713426562764215319, 11.30455417841141948737084501739

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