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Properties

Label 2-684-76.75-c1-0-25
Degree $2$
Conductor $684$
Sign $-0.603 + 0.797i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.603 + 0.797i$
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: \((2,\ 684,\ (\ :1/2),\ -0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284464 - 0.571988i\)
\(L(\frac12)\) \(\approx\) \(0.284464 - 0.571988i\)
\(L(\frac{3}{2})\) 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

−10.24869675299735212877112727998, −9.500455956677087192933268154539, −8.376287809918084997579562627548, −7.68987311117922471235949876635, −7.13023465492891773857182006677, −5.63352655349835856227096793302, −4.24481875544345471999720438801, −3.46328319864834308105022103474, −2.13900389619513396126955372082, −0.46253324461871902463256098240, 1.41016452664011961017725154764, 3.13537410747806583972415036547, 4.53308519938691901836643627936, 5.57273986664828842609470686204, 6.52421743076148452953279652310, 7.40542830527716612373126845946, 8.139843177557715544819212478001, 9.121456173122636837354447250535, 9.507038410134808945850778244388, 10.91148944603107802045919782213

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