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Properties

Label 2-684-76.75-c1-0-46
Degree $2$
Conductor $684$
Sign $-0.801 + 0.598i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.06 − 0.927i)2-s + (0.280 − 1.98i)4-s − 1.56·5-s + 0.868i·7-s + (−1.53 − 2.37i)8-s + (−1.66 + 1.44i)10-s − 3.09i·11-s − 4.74i·13-s + (0.804 + 0.927i)14-s + (−3.84 − 1.11i)16-s + 17-s + (−3.07 − 3.09i)19-s + (−0.438 + 3.09i)20-s + (−2.86 − 3.30i)22-s + 3.96i·23-s + ⋯
L(s)  = 1  + (0.755 − 0.655i)2-s + (0.140 − 0.990i)4-s − 0.698·5-s + 0.328i·7-s + (−0.543 − 0.839i)8-s + (−0.527 + 0.457i)10-s − 0.932i·11-s − 1.31i·13-s + (0.215 + 0.247i)14-s + (−0.960 − 0.277i)16-s + 0.242·17-s + (−0.704 − 0.709i)19-s + (−0.0980 + 0.691i)20-s + (−0.611 − 0.704i)22-s + 0.825i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.801 + 0.598i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.801 + 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509859 - 1.53519i\)
\(L(\frac12)\) \(\approx\) \(0.509859 - 1.53519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.927i)T \)
3 \( 1 \)
19 \( 1 + (3.07 + 3.09i)T \)
good5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 - 0.868iT - 7T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 + 4.74iT - 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
23 \( 1 - 3.96iT - 23T^{2} \)
29 \( 1 + 8.45iT - 29T^{2} \)
31 \( 1 - 4.27T + 31T^{2} \)
37 \( 1 + 3.70iT - 37T^{2} \)
41 \( 1 + 3.70iT - 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 - 9.27iT - 47T^{2} \)
53 \( 1 - 1.04iT - 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 0.684T + 61T^{2} \)
67 \( 1 - 9.74T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 + 9.65iT - 83T^{2} \)
89 \( 1 + 5.79iT - 89T^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−10.36375361339933507105751845039, −9.480148080389988242546064143216, −8.390562747735111089634672815463, −7.57287992486477194599092410862, −6.19745707331593499308583651107, −5.57793646994069736022458151239, −4.41960701693332271066681314475, −3.45475514621217890845690843478, −2.52524649555806636278711225686, −0.66209156258401067109058777418, 2.14678743864568302222825788861, 3.72372802209407628753279825620, 4.31432844648699166210013828276, 5.28205909625484808779839208172, 6.66087037483288040998162335820, 7.03058736096028131698132115942, 8.089182597456508512600879768391, 8.787570263079412449342170708339, 9.981811991131365662062989797725, 10.99258520048763220775426443676

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