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Properties

Label 2-684-19.18-c6-0-42
Degree $2$
Conductor $684$
Sign $-0.364 + 0.931i$
Analytic cond. $157.356$
Root an. cond. $12.5442$
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  + 18.1·5-s + 85.5·7-s + 316.·11-s − 2.45e3i·13-s + 3.22e3·17-s + (2.50e3 − 6.38e3i)19-s + 7.82e3·23-s − 1.52e4·25-s + 1.89e4i·29-s − 3.23e4i·31-s + 1.54e3·35-s − 1.54e4i·37-s + 9.41e4i·41-s + 3.92e4·43-s − 9.23e4·47-s + ⋯
L(s)  = 1  + 0.144·5-s + 0.249·7-s + 0.237·11-s − 1.11i·13-s + 0.656·17-s + (0.364 − 0.931i)19-s + 0.643·23-s − 0.978·25-s + 0.778i·29-s − 1.08i·31-s + 0.0361·35-s − 0.304i·37-s + 1.36i·41-s + 0.493·43-s − 0.889·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.364 + 0.931i$
Analytic conductor: \(157.356\)
Root analytic conductor: \(12.5442\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :3),\ -0.364 + 0.931i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.760439370\)
\(L(\frac12)\) \(\approx\) \(1.760439370\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-2.50e3 + 6.38e3i)T \)
good5 \( 1 - 18.1T + 1.56e4T^{2} \)
7 \( 1 - 85.5T + 1.17e5T^{2} \)
11 \( 1 - 316.T + 1.77e6T^{2} \)
13 \( 1 + 2.45e3iT - 4.82e6T^{2} \)
17 \( 1 - 3.22e3T + 2.41e7T^{2} \)
23 \( 1 - 7.82e3T + 1.48e8T^{2} \)
29 \( 1 - 1.89e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.23e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.54e4iT - 2.56e9T^{2} \)
41 \( 1 - 9.41e4iT - 4.75e9T^{2} \)
43 \( 1 - 3.92e4T + 6.32e9T^{2} \)
47 \( 1 + 9.23e4T + 1.07e10T^{2} \)
53 \( 1 + 1.30e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.80e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.24e5T + 5.15e10T^{2} \)
67 \( 1 - 1.54e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.77e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.14e5T + 1.51e11T^{2} \)
79 \( 1 - 4.62e5iT - 2.43e11T^{2} \)
83 \( 1 + 6.01e5T + 3.26e11T^{2} \)
89 \( 1 - 2.75e5iT - 4.96e11T^{2} \)
97 \( 1 + 8.52e5iT - 8.32e11T^{2} \)
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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

−9.415783302797542698566197026085, −8.295007981674268387829954637502, −7.64263308116433977836533617471, −6.62444092706539118474609262271, −5.60137760323773215367666016917, −4.86018769853600214580313665002, −3.59991628834572217738613120636, −2.67727231629582479397862010935, −1.38506221763300751132095995121, −0.35005643109908448609229899077, 1.15531982921960492646704506006, 2.03797709366526439547678729036, 3.37056166652689710425808305754, 4.30223464545563210264354602135, 5.35915236404642166519303395725, 6.25979875955972458621732367304, 7.21110945317030393236208855112, 8.062764542828448438443325207623, 9.020526892036217939517234881986, 9.760728057424006197679819925018

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