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Properties

Label 2-684-4.3-c2-0-28
Degree $2$
Conductor $684$
Sign $0.157 + 0.987i$
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  + (−0.157 − 1.99i)2-s + (−3.95 + 0.630i)4-s − 9.57·5-s − 3.50i·7-s + (1.88 + 7.77i)8-s + (1.51 + 19.0i)10-s + 18.8i·11-s − 0.861·13-s + (−6.98 + 0.553i)14-s + (15.2 − 4.97i)16-s − 7.12·17-s + 4.35i·19-s + (37.8 − 6.03i)20-s + (37.5 − 2.97i)22-s − 31.5i·23-s + ⋯
L(s)  = 1  + (−0.0789 − 0.996i)2-s + (−0.987 + 0.157i)4-s − 1.91·5-s − 0.500i·7-s + (0.235 + 0.971i)8-s + (0.151 + 1.90i)10-s + 1.71i·11-s − 0.0662·13-s + (−0.498 + 0.0395i)14-s + (0.950 − 0.311i)16-s − 0.419·17-s + 0.229i·19-s + (1.89 − 0.301i)20-s + (1.70 − 0.135i)22-s − 1.37i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.157 + 0.987i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.157 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7209965104\)
\(L(\frac12)\) \(\approx\) \(0.7209965104\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.157 + 1.99i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 9.57T + 25T^{2} \)
7 \( 1 + 3.50iT - 49T^{2} \)
11 \( 1 - 18.8iT - 121T^{2} \)
13 \( 1 + 0.861T + 169T^{2} \)
17 \( 1 + 7.12T + 289T^{2} \)
23 \( 1 + 31.5iT - 529T^{2} \)
29 \( 1 + 31.6T + 841T^{2} \)
31 \( 1 + 5.98iT - 961T^{2} \)
37 \( 1 + 39.8T + 1.36e3T^{2} \)
41 \( 1 - 23.4T + 1.68e3T^{2} \)
43 \( 1 - 10.4iT - 1.84e3T^{2} \)
47 \( 1 + 33.5iT - 2.20e3T^{2} \)
53 \( 1 - 88.3T + 2.80e3T^{2} \)
59 \( 1 + 23.6iT - 3.48e3T^{2} \)
61 \( 1 + 53.6T + 3.72e3T^{2} \)
67 \( 1 + 89.8iT - 4.48e3T^{2} \)
71 \( 1 + 16.1iT - 5.04e3T^{2} \)
73 \( 1 - 126.T + 5.32e3T^{2} \)
79 \( 1 - 127. iT - 6.24e3T^{2} \)
83 \( 1 - 59.0iT - 6.88e3T^{2} \)
89 \( 1 - 108.T + 7.92e3T^{2} \)
97 \( 1 - 38.0T + 9.40e3T^{2} \)
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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

−10.37498848458871886610918573381, −9.298070443845462095661223181228, −8.382858792219852921905255332000, −7.58525965439833610364090807634, −6.93227082608478174057724161113, −4.94160486004731641690626220022, −4.25741364344884161753492531757, −3.61867296399510514979484911109, −2.19977360519569373215709190503, −0.51026154447053996267215157175, 0.63848903884251663335678877145, 3.30380052789384738827610731121, 3.95220337703252038452100689002, 5.14112150494847043681787203969, 6.03085019199557943336143952662, 7.21894425335964784790537364115, 7.75453324030364592171495516185, 8.708850936454164772730479628966, 8.997983137857895190493070112867, 10.58358895257208307228693193289

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