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Properties

Label 2-684-228.227-c2-0-43
Degree $2$
Conductor $684$
Sign $0.972 + 0.233i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 − 1.22i)2-s + (1.01 + 3.86i)4-s + 8.58i·5-s − 1.97i·7-s + (3.11 − 7.36i)8-s + (10.4 − 13.5i)10-s + 19.5·11-s − 18.4i·13-s + (−2.41 + 3.12i)14-s + (−13.9 + 7.86i)16-s − 22.2i·17-s + (−18.8 + 2.33i)19-s + (−33.1 + 8.72i)20-s + (−30.9 − 23.8i)22-s + 15.3·23-s + ⋯
L(s)  = 1  + (−0.791 − 0.610i)2-s + (0.254 + 0.967i)4-s + 1.71i·5-s − 0.282i·7-s + (0.389 − 0.921i)8-s + (1.04 − 1.35i)10-s + 1.77·11-s − 1.41i·13-s + (−0.172 + 0.223i)14-s + (−0.870 + 0.491i)16-s − 1.30i·17-s + (−0.992 + 0.122i)19-s + (−1.65 + 0.436i)20-s + (−1.40 − 1.08i)22-s + 0.665·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.972 + 0.233i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.972 + 0.233i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.357220284\)
\(L(\frac12)\) \(\approx\) \(1.357220284\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.58 + 1.22i)T \)
3 \( 1 \)
19 \( 1 + (18.8 - 2.33i)T \)
good5 \( 1 - 8.58iT - 25T^{2} \)
7 \( 1 + 1.97iT - 49T^{2} \)
11 \( 1 - 19.5T + 121T^{2} \)
13 \( 1 + 18.4iT - 169T^{2} \)
17 \( 1 + 22.2iT - 289T^{2} \)
23 \( 1 - 15.3T + 529T^{2} \)
29 \( 1 - 38.6T + 841T^{2} \)
31 \( 1 - 45.2T + 961T^{2} \)
37 \( 1 + 9.92iT - 1.36e3T^{2} \)
41 \( 1 - 36.5T + 1.68e3T^{2} \)
43 \( 1 + 25.5iT - 1.84e3T^{2} \)
47 \( 1 + 6.62T + 2.20e3T^{2} \)
53 \( 1 + 68.3T + 2.80e3T^{2} \)
59 \( 1 - 83.5iT - 3.48e3T^{2} \)
61 \( 1 - 14.9T + 3.72e3T^{2} \)
67 \( 1 + 92.3T + 4.48e3T^{2} \)
71 \( 1 - 43.3iT - 5.04e3T^{2} \)
73 \( 1 - 45.8T + 5.32e3T^{2} \)
79 \( 1 - 84.8T + 6.24e3T^{2} \)
83 \( 1 - 11.0T + 6.88e3T^{2} \)
89 \( 1 + 58.4T + 7.92e3T^{2} \)
97 \( 1 + 30.6iT - 9.40e3T^{2} \)
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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

−10.35947305438729592897955422731, −9.606879770204624921702385093262, −8.629113117714386225670401672653, −7.58155549374089292188902126551, −6.86467884290397236066626339350, −6.23206533100489447122279478956, −4.30937233992972043515772555500, −3.23544982555357658139777410171, −2.56753606366817702916433636216, −0.866422975781856968221093352743, 1.01958673975196651403667270564, 1.81917904471599038169828912957, 4.23737093793701942103076670686, 4.75531945032698483343069821221, 6.22209327844879734790398303828, 6.53477879091534283387624029497, 8.046004513205382628658529307183, 8.782593244586457034539005894924, 9.077238953621674353031277361175, 9.900239628921217179367696527210

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