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Properties

Label 2-684-4.3-c2-0-76
Degree $2$
Conductor $684$
Sign $-0.357 - 0.934i$
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  + (−0.363 − 1.96i)2-s + (−3.73 + 1.42i)4-s − 0.0632·5-s − 2.71i·7-s + (4.16 + 6.82i)8-s + (0.0229 + 0.124i)10-s − 1.99i·11-s + 9.23·13-s + (−5.34 + 0.987i)14-s + (11.9 − 10.6i)16-s − 29.7·17-s + 4.35i·19-s + (0.236 − 0.0904i)20-s + (−3.91 + 0.723i)22-s − 30.0i·23-s + ⋯
L(s)  = 1  + (−0.181 − 0.983i)2-s + (−0.934 + 0.357i)4-s − 0.0126·5-s − 0.388i·7-s + (0.520 + 0.853i)8-s + (0.00229 + 0.0124i)10-s − 0.180i·11-s + 0.710·13-s + (−0.381 + 0.0705i)14-s + (0.744 − 0.667i)16-s − 1.75·17-s + 0.229i·19-s + (0.0118 − 0.00452i)20-s + (−0.177 + 0.0328i)22-s − 1.30i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.357 - 0.934i$
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: \((2,\ 684,\ (\ :1),\ -0.357 - 0.934i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05294248016\)
\(L(\frac12)\) \(\approx\) \(0.05294248016\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.363 + 1.96i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 0.0632T + 25T^{2} \)
7 \( 1 + 2.71iT - 49T^{2} \)
11 \( 1 + 1.99iT - 121T^{2} \)
13 \( 1 - 9.23T + 169T^{2} \)
17 \( 1 + 29.7T + 289T^{2} \)
23 \( 1 + 30.0iT - 529T^{2} \)
29 \( 1 - 1.04T + 841T^{2} \)
31 \( 1 - 24.8iT - 961T^{2} \)
37 \( 1 + 22.4T + 1.36e3T^{2} \)
41 \( 1 + 32.4T + 1.68e3T^{2} \)
43 \( 1 - 48.6iT - 1.84e3T^{2} \)
47 \( 1 + 13.7iT - 2.20e3T^{2} \)
53 \( 1 + 87.2T + 2.80e3T^{2} \)
59 \( 1 - 74.0iT - 3.48e3T^{2} \)
61 \( 1 + 31.9T + 3.72e3T^{2} \)
67 \( 1 - 115. iT - 4.48e3T^{2} \)
71 \( 1 + 66.3iT - 5.04e3T^{2} \)
73 \( 1 + 74.8T + 5.32e3T^{2} \)
79 \( 1 - 45.5iT - 6.24e3T^{2} \)
83 \( 1 - 4.33iT - 6.88e3T^{2} \)
89 \( 1 + 111.T + 7.92e3T^{2} \)
97 \( 1 + 85.9T + 9.40e3T^{2} \)
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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

−9.854805649710397706514010350184, −8.795020342918059480469414927944, −8.357296317020132062914765338716, −7.10937600031289069546535680533, −6.04283942195111040790729841443, −4.69513388101252972483497206879, −3.96597417524861786480511310452, −2.77722799663516420933693405599, −1.53995981577732158757522937662, −0.02015723149158042947614319456, 1.84414021339971282367308492291, 3.63416120103203350630395377151, 4.64896527758861260660677489517, 5.67514352515298334733637899129, 6.45902188904676779449033327924, 7.34181265182750042912766648275, 8.275587021875063856656505898909, 9.037011534225902874221231430992, 9.684990774592570033265694264716, 10.79548015008918395618739364804

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