Properties

Label 2-304-19.18-c6-0-53
Degree $2$
Conductor $304$
Sign $0.388 + 0.921i$
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  − 17.1i·3-s + 166.·5-s + 600.·7-s + 434.·9-s − 821.·11-s − 3.12e3i·13-s − 2.86e3i·15-s + 6.60e3·17-s + (−2.66e3 − 6.32e3i)19-s − 1.02e4i·21-s + 3.41e3·23-s + 1.22e4·25-s − 1.99e4i·27-s − 3.63e4i·29-s + 3.83e4i·31-s + ⋯
L(s)  = 1  − 0.635i·3-s + 1.33·5-s + 1.75·7-s + 0.596·9-s − 0.616·11-s − 1.42i·13-s − 0.848i·15-s + 1.34·17-s + (−0.388 − 0.921i)19-s − 1.11i·21-s + 0.280·23-s + 0.784·25-s − 1.01i·27-s − 1.49i·29-s + 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.388 + 0.921i$
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: \((2,\ 304,\ (\ :3),\ 0.388 + 0.921i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.852545308\)
\(L(\frac12)\) \(\approx\) \(3.852545308\)
\(L(4)\) 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 \( 1 + (2.66e3 + 6.32e3i)T \)
good3 \( 1 + 17.1iT - 729T^{2} \)
5 \( 1 - 166.T + 1.56e4T^{2} \)
7 \( 1 - 600.T + 1.17e5T^{2} \)
11 \( 1 + 821.T + 1.77e6T^{2} \)
13 \( 1 + 3.12e3iT - 4.82e6T^{2} \)
17 \( 1 - 6.60e3T + 2.41e7T^{2} \)
23 \( 1 - 3.41e3T + 1.48e8T^{2} \)
29 \( 1 + 3.63e4iT - 5.94e8T^{2} \)
31 \( 1 - 3.83e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.14e4iT - 2.56e9T^{2} \)
41 \( 1 - 7.66e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.28e5T + 6.32e9T^{2} \)
47 \( 1 + 1.00e5T + 1.07e10T^{2} \)
53 \( 1 - 1.51e5iT - 2.21e10T^{2} \)
59 \( 1 + 8.63e4iT - 4.21e10T^{2} \)
61 \( 1 - 1.48e5T + 5.15e10T^{2} \)
67 \( 1 + 3.36e5iT - 9.04e10T^{2} \)
71 \( 1 - 6.80e4iT - 1.28e11T^{2} \)
73 \( 1 - 4.79e5T + 1.51e11T^{2} \)
79 \( 1 - 7.13e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.50e5T + 3.26e11T^{2} \)
89 \( 1 + 4.59e5iT - 4.96e11T^{2} \)
97 \( 1 + 3.48e5iT - 8.32e11T^{2} \)
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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

−10.37773441285563785432055137144, −9.825145753310576950152787005460, −8.242978576590373318130780831056, −7.85909764975728223355696802398, −6.58659188074981069870619274654, −5.41342944922625121052420744830, −4.83569234499008970373670804319, −2.82109163208456474767689119446, −1.71091968735778893312148426780, −0.982592117501664767587316841423, 1.47958124785640788940539408416, 2.00841267695890335547254485666, 3.88181195930792655158055907706, 4.99439112123073673744342965829, 5.57933895964169255219486159886, 7.01600235819030042609416345940, 8.096475662313571198494937212176, 9.147539580604141383167461369703, 9.999007367650930243704513839023, 10.66492030593117238551864979595

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