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Properties

Label 2-684-19.12-c4-0-6
Degree $2$
Conductor $684$
Sign $-0.896 - 0.443i$
Analytic cond. $70.7050$
Root an. cond. $8.40862$
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  + (22.6 + 39.1i)5-s + 80.9·7-s − 74.1·11-s + (−249. − 144. i)13-s + (255. + 441. i)17-s + (−356. + 58.3i)19-s + (−122. + 212. i)23-s + (−709. + 1.22e3i)25-s + (−932. − 538. i)29-s − 96.0i·31-s + (1.82e3 + 3.16e3i)35-s + 1.33e3i·37-s + (−1.75e3 + 1.01e3i)41-s + (−1.40e3 − 2.43e3i)43-s + (−1.40e3 + 2.44e3i)47-s + ⋯
L(s)  = 1  + (0.904 + 1.56i)5-s + 1.65·7-s − 0.612·11-s + (−1.47 − 0.853i)13-s + (0.882 + 1.52i)17-s + (−0.986 + 0.161i)19-s + (−0.231 + 0.400i)23-s + (−1.13 + 1.96i)25-s + (−1.10 − 0.640i)29-s − 0.0999i·31-s + (1.49 + 2.58i)35-s + 0.975i·37-s + (−1.04 + 0.602i)41-s + (−0.761 − 1.31i)43-s + (−0.637 + 1.10i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.896 - 0.443i$
Analytic conductor: \(70.7050\)
Root analytic conductor: \(8.40862\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :2),\ -0.896 - 0.443i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.792077199\)
\(L(\frac12)\) \(\approx\) \(1.792077199\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (356. - 58.3i)T \)
good5 \( 1 + (-22.6 - 39.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 80.9T + 2.40e3T^{2} \)
11 \( 1 + 74.1T + 1.46e4T^{2} \)
13 \( 1 + (249. + 144. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-255. - 441. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (122. - 212. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (932. + 538. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + 96.0iT - 9.23e5T^{2} \)
37 \( 1 - 1.33e3iT - 1.87e6T^{2} \)
41 \( 1 + (1.75e3 - 1.01e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.40e3 + 2.43e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.40e3 - 2.44e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (754. + 435. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-5.31e3 + 3.06e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (14.3 - 24.8i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.61e3 - 2.08e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-2.89e3 + 1.66e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (457. + 791. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-413. + 238. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 1.08e3T + 4.74e7T^{2} \)
89 \( 1 + (-1.05e4 - 6.06e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (1.04e4 - 6.04e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.29499989979461433887218305436, −9.807763431360850631244277517542, −8.144301205577542861683082708590, −7.82017456052411979349858474403, −6.75398571426276537698734058273, −5.71116907384853651226914445029, −5.04944642764162634671138819210, −3.59515089605257528430082454530, −2.37196773778207662598172789428, −1.74766245175062967055377457339, 0.37129104928632598821714561007, 1.65010353780972695200287765753, 2.30316766600947235064486231794, 4.38877796064607685364838133930, 5.11586983880201640663939359631, 5.36479437104570855158004216204, 7.02308201405146969693848371846, 7.944564727905865500698043542345, 8.693132371887744810170280260290, 9.458067441879460231739411467315

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