Properties

Label 2-684-228.227-c2-0-53
Degree $2$
Conductor $684$
Sign $-0.702 + 0.711i$
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  + (−1.90 − 0.613i)2-s + (3.24 + 2.33i)4-s + 6.64i·5-s + 0.470i·7-s + (−4.74 − 6.43i)8-s + (4.07 − 12.6i)10-s − 10.1·11-s − 3.45i·13-s + (0.288 − 0.896i)14-s + (5.09 + 15.1i)16-s + 14.2i·17-s + (−13.6 − 13.2i)19-s + (−15.5 + 21.5i)20-s + (19.3 + 6.24i)22-s − 25.5·23-s + ⋯
L(s)  = 1  + (−0.951 − 0.306i)2-s + (0.811 + 0.583i)4-s + 1.32i·5-s + 0.0672i·7-s + (−0.593 − 0.804i)8-s + (0.407 − 1.26i)10-s − 0.924·11-s − 0.265i·13-s + (0.0206 − 0.0640i)14-s + (0.318 + 0.948i)16-s + 0.839i·17-s + (−0.717 − 0.696i)19-s + (−0.775 + 1.07i)20-s + (0.880 + 0.283i)22-s − 1.11·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.702 + 0.711i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1213039496\)
\(L(\frac12)\) \(\approx\) \(0.1213039496\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.90 + 0.613i)T \)
3 \( 1 \)
19 \( 1 + (13.6 + 13.2i)T \)
good5 \( 1 - 6.64iT - 25T^{2} \)
7 \( 1 - 0.470iT - 49T^{2} \)
11 \( 1 + 10.1T + 121T^{2} \)
13 \( 1 + 3.45iT - 169T^{2} \)
17 \( 1 - 14.2iT - 289T^{2} \)
23 \( 1 + 25.5T + 529T^{2} \)
29 \( 1 + 14.1T + 841T^{2} \)
31 \( 1 - 19.3T + 961T^{2} \)
37 \( 1 + 30.0iT - 1.36e3T^{2} \)
41 \( 1 - 25.6T + 1.68e3T^{2} \)
43 \( 1 + 38.5iT - 1.84e3T^{2} \)
47 \( 1 - 16.5T + 2.20e3T^{2} \)
53 \( 1 + 41.6T + 2.80e3T^{2} \)
59 \( 1 + 31.3iT - 3.48e3T^{2} \)
61 \( 1 + 3.53T + 3.72e3T^{2} \)
67 \( 1 - 13.9T + 4.48e3T^{2} \)
71 \( 1 + 128. iT - 5.04e3T^{2} \)
73 \( 1 + 71.6T + 5.32e3T^{2} \)
79 \( 1 + 100.T + 6.24e3T^{2} \)
83 \( 1 - 58.6T + 6.88e3T^{2} \)
89 \( 1 + 108.T + 7.92e3T^{2} \)
97 \( 1 - 124. iT - 9.40e3T^{2} \)
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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

−10.26946248288066031916871881736, −9.163290697868653793380891166668, −8.158169815538998812058828759547, −7.48924964280374038195355393083, −6.60746974838572191247371550257, −5.78338848230684648982574992832, −4.00573745881317192903640879798, −2.88529688228188133419374331427, −2.08701194120030859651984597191, −0.06028258608397030125324523799, 1.25910362799159007493574612098, 2.54082880135795141338590992974, 4.34375471168127229665592324417, 5.33384563974289574062864040806, 6.15789107224442480401920820790, 7.39895726496422776519881342008, 8.158107367708437671736154992626, 8.764561454276660643497551293146, 9.671241365581507921254668518967, 10.29298112665637107804935570114

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