Properties

Label 2-684-171.103-c2-0-34
Degree $2$
Conductor $684$
Sign $-0.877 - 0.480i$
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  + (−2.67 + 1.36i)3-s + (−4.45 − 7.71i)5-s + (−2.28 − 3.95i)7-s + (5.27 − 7.29i)9-s + (−0.526 − 0.912i)11-s − 16.8i·13-s + (22.4 + 14.5i)15-s + (3.48 − 6.03i)17-s + (17.6 − 6.97i)19-s + (11.4 + 7.44i)21-s − 8.67·23-s + (−27.1 + 47.0i)25-s + (−4.12 + 26.6i)27-s + (−32.0 − 18.4i)29-s + (−9.57 − 5.52i)31-s + ⋯
L(s)  = 1  + (−0.890 + 0.455i)3-s + (−0.890 − 1.54i)5-s + (−0.325 − 0.564i)7-s + (0.585 − 0.810i)9-s + (−0.0478 − 0.0829i)11-s − 1.29i·13-s + (1.49 + 0.968i)15-s + (0.205 − 0.355i)17-s + (0.930 − 0.367i)19-s + (0.547 + 0.354i)21-s − 0.377·23-s + (−1.08 + 1.88i)25-s + (−0.152 + 0.988i)27-s + (−1.10 − 0.637i)29-s + (−0.308 − 0.178i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.877 - 0.480i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.877 - 0.480i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4131555903\)
\(L(\frac12)\) \(\approx\) \(0.4131555903\)
\(L(2)\) 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 \( 1 + (2.67 - 1.36i)T \)
19 \( 1 + (-17.6 + 6.97i)T \)
good5 \( 1 + (4.45 + 7.71i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.28 + 3.95i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.526 + 0.912i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 16.8iT - 169T^{2} \)
17 \( 1 + (-3.48 + 6.03i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 8.67T + 529T^{2} \)
29 \( 1 + (32.0 + 18.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (9.57 + 5.52i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 11.0iT - 1.36e3T^{2} \)
41 \( 1 + (18.4 - 10.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 51.1T + 1.84e3T^{2} \)
47 \( 1 + (13.8 - 24.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (11.4 - 6.60i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (23.2 - 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.965 - 1.67i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 23.9iT - 4.48e3T^{2} \)
71 \( 1 + (102. + 59.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (46.1 - 79.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 38.4iT - 6.24e3T^{2} \)
83 \( 1 + (-36.8 - 63.7i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-86.6 + 50.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 143. iT - 9.40e3T^{2} \)
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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

−9.740814663669552298249981323382, −9.086354601117945950100269148646, −7.922931640423939887190938427574, −7.32607682597840935486952100289, −5.83646452603304871401573085711, −5.18503410376815978670278762085, −4.29229934026291258235487227402, −3.44161241610125924092342635619, −1.00063367064746128314081713172, −0.21007867025734715101343272477, 1.89407042182311561437172164595, 3.21037758366307137387316743928, 4.25676955373845290047479411997, 5.64514328741389626328092029170, 6.45235604552743118215288267840, 7.19825411673563956478669942439, 7.76662681843118298263794447366, 9.130510745278583730654794805726, 10.21516721603882373576049167848, 10.88752725891221979277548646665

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