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Properties

Label 2-684-4.3-c2-0-56
Degree $2$
Conductor $684$
Sign $0.204 + 0.978i$
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  + (−1.98 + 0.205i)2-s + (3.91 − 0.817i)4-s − 0.0956·5-s − 9.56i·7-s + (−7.62 + 2.43i)8-s + (0.190 − 0.0196i)10-s + 7.63i·11-s + 7.83·13-s + (1.96 + 19.0i)14-s + (14.6 − 6.40i)16-s + 16.6·17-s + 4.35i·19-s + (−0.374 + 0.0781i)20-s + (−1.56 − 15.1i)22-s − 6.37i·23-s + ⋯
L(s)  = 1  + (−0.994 + 0.102i)2-s + (0.978 − 0.204i)4-s − 0.0191·5-s − 1.36i·7-s + (−0.952 + 0.303i)8-s + (0.0190 − 0.00196i)10-s + 0.693i·11-s + 0.603·13-s + (0.140 + 1.35i)14-s + (0.916 − 0.400i)16-s + 0.982·17-s + 0.229i·19-s + (−0.0187 + 0.00390i)20-s + (−0.0712 − 0.690i)22-s − 0.277i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.204 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.023233753\)
\(L(\frac12)\) \(\approx\) \(1.023233753\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 - 0.205i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 0.0956T + 25T^{2} \)
7 \( 1 + 9.56iT - 49T^{2} \)
11 \( 1 - 7.63iT - 121T^{2} \)
13 \( 1 - 7.83T + 169T^{2} \)
17 \( 1 - 16.6T + 289T^{2} \)
23 \( 1 + 6.37iT - 529T^{2} \)
29 \( 1 - 19.5T + 841T^{2} \)
31 \( 1 + 57.7iT - 961T^{2} \)
37 \( 1 - 50.5T + 1.36e3T^{2} \)
41 \( 1 + 57.9T + 1.68e3T^{2} \)
43 \( 1 - 13.2iT - 1.84e3T^{2} \)
47 \( 1 - 42.0iT - 2.20e3T^{2} \)
53 \( 1 + 40.5T + 2.80e3T^{2} \)
59 \( 1 + 89.4iT - 3.48e3T^{2} \)
61 \( 1 - 30.2T + 3.72e3T^{2} \)
67 \( 1 + 61.0iT - 4.48e3T^{2} \)
71 \( 1 + 134. iT - 5.04e3T^{2} \)
73 \( 1 - 50.8T + 5.32e3T^{2} \)
79 \( 1 + 10.4iT - 6.24e3T^{2} \)
83 \( 1 + 80.3iT - 6.88e3T^{2} \)
89 \( 1 + 73.9T + 7.92e3T^{2} \)
97 \( 1 + 101.T + 9.40e3T^{2} \)
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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

−9.906057710734847899020178014605, −9.535193507365695817726616402237, −8.036183065062089407937047195924, −7.77281568934023889082977529508, −6.73880214839123435856267963813, −5.91878039983275541142488572136, −4.45072759059568679681777131145, −3.33634569933104455607305562744, −1.77967775583981967794224323652, −0.56833100657302316903414425643, 1.23445602949793963149367551503, 2.57546087069843586641952976799, 3.51459538727847236589195646308, 5.38529910226859943368820417533, 6.04661783624104867322381459691, 7.05589137000597204211046579839, 8.316267724903822148199722752225, 8.562753146542087583303926809430, 9.553869363147138904293699738482, 10.30830557627414699871795538671

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