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Properties

Label 2-684-19.12-c2-0-2
Degree $2$
Conductor $684$
Sign $-0.184 - 0.982i$
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  + (−3.36 − 5.82i)5-s − 10.3·7-s + 11.0·11-s + (2.58 + 1.49i)13-s + (−3.79 − 6.57i)17-s + (−13.7 − 13.1i)19-s + (−6.15 + 10.6i)23-s + (−10.1 + 17.5i)25-s + (−0.256 − 0.148i)29-s + 43.6i·31-s + (34.7 + 60.1i)35-s + 36.3i·37-s + (−21.8 + 12.6i)41-s + (25.2 + 43.6i)43-s + (−31.4 + 54.4i)47-s + ⋯
L(s)  = 1  + (−0.672 − 1.16i)5-s − 1.47·7-s + 1.00·11-s + (0.198 + 0.114i)13-s + (−0.223 − 0.386i)17-s + (−0.722 − 0.691i)19-s + (−0.267 + 0.463i)23-s + (−0.405 + 0.702i)25-s + (−0.00885 − 0.00511i)29-s + 1.40i·31-s + (0.992 + 1.71i)35-s + 0.983i·37-s + (−0.533 + 0.308i)41-s + (0.586 + 1.01i)43-s + (−0.669 + 1.15i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.184 - 0.982i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4045443838\)
\(L(\frac12)\) \(\approx\) \(0.4045443838\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (13.7 + 13.1i)T \)
good5 \( 1 + (3.36 + 5.82i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 - 11.0T + 121T^{2} \)
13 \( 1 + (-2.58 - 1.49i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (3.79 + 6.57i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (6.15 - 10.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (0.256 + 0.148i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 43.6iT - 961T^{2} \)
37 \( 1 - 36.3iT - 1.36e3T^{2} \)
41 \( 1 + (21.8 - 12.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25.2 - 43.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (31.4 - 54.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-49.5 - 28.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (53.8 - 31.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-17.4 + 30.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.8 - 16.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (12.9 - 7.48i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (30.6 + 53.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-0.933 + 0.539i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 128.T + 6.88e3T^{2} \)
89 \( 1 + (73.8 + 42.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-24.9 + 14.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.43995112759472875100588411746, −9.325358594083110472196082748367, −9.038579357162937209359241979828, −8.044416942302472576060307182746, −6.84675082646027201344741457935, −6.23353213916238834917402948851, −4.88682028757543381428793334156, −4.04452126249383853203867630673, −3.03198007724421813397458871232, −1.17471199589354271863438141272, 0.16092004514940260604991598452, 2.31699853554778535464004370922, 3.59356277569641134185245383218, 3.95322740510763020126806403799, 5.87849352626019659962351458847, 6.56596854848799694910883664423, 7.15901707896475103884778137408, 8.291493338083565428114409466538, 9.274295252911901747188510265814, 10.13895535621383059262427185480

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