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Properties

Label 2-684-4.3-c2-0-89
Degree $2$
Conductor $684$
Sign $0.647 - 0.761i$
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  + (0.690 − 1.87i)2-s + (−3.04 − 2.59i)4-s − 8.91·5-s − 7.74i·7-s + (−6.96 + 3.93i)8-s + (−6.15 + 16.7i)10-s − 16.0i·11-s − 6.44·13-s + (−14.5 − 5.34i)14-s + (2.57 + 15.7i)16-s + 1.80·17-s + 4.35i·19-s + (27.1 + 23.0i)20-s + (−30.1 − 11.0i)22-s + 9.27i·23-s + ⋯
L(s)  = 1  + (0.345 − 0.938i)2-s + (−0.761 − 0.647i)4-s − 1.78·5-s − 1.10i·7-s + (−0.870 + 0.491i)8-s + (−0.615 + 1.67i)10-s − 1.46i·11-s − 0.495·13-s + (−1.03 − 0.381i)14-s + (0.160 + 0.987i)16-s + 0.106·17-s + 0.229i·19-s + (1.35 + 1.15i)20-s + (−1.37 − 0.504i)22-s + 0.403i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.647 - 0.761i$
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: \((2,\ 684,\ (\ :1),\ 0.647 - 0.761i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01111470617\)
\(L(\frac12)\) \(\approx\) \(0.01111470617\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.690 + 1.87i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 8.91T + 25T^{2} \)
7 \( 1 + 7.74iT - 49T^{2} \)
11 \( 1 + 16.0iT - 121T^{2} \)
13 \( 1 + 6.44T + 169T^{2} \)
17 \( 1 - 1.80T + 289T^{2} \)
23 \( 1 - 9.27iT - 529T^{2} \)
29 \( 1 + 40.7T + 841T^{2} \)
31 \( 1 - 25.0iT - 961T^{2} \)
37 \( 1 - 71.0T + 1.36e3T^{2} \)
41 \( 1 + 49.0T + 1.68e3T^{2} \)
43 \( 1 + 30.1iT - 1.84e3T^{2} \)
47 \( 1 - 82.6iT - 2.20e3T^{2} \)
53 \( 1 - 35.2T + 2.80e3T^{2} \)
59 \( 1 + 80.2iT - 3.48e3T^{2} \)
61 \( 1 - 51.2T + 3.72e3T^{2} \)
67 \( 1 - 4.89iT - 4.48e3T^{2} \)
71 \( 1 - 96.5iT - 5.04e3T^{2} \)
73 \( 1 + 117.T + 5.32e3T^{2} \)
79 \( 1 + 6.35iT - 6.24e3T^{2} \)
83 \( 1 + 91.5iT - 6.88e3T^{2} \)
89 \( 1 + 87.1T + 7.92e3T^{2} \)
97 \( 1 + 107.T + 9.40e3T^{2} \)
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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

−9.749978920876883501175529959804, −8.619479339966041173590738705621, −7.914237788453618550118949900074, −7.03633809724237951398998801238, −5.64936811392171346794714233309, −4.44676291531486090448243552073, −3.76586111575458098961442723057, −3.06838492170945075472008719067, −1.00849279313944786670682197168, −0.00450353367911300090984222676, 2.61361434910454023151519784022, 3.92879570625801789403503763612, 4.61349314116135441272205672225, 5.56124044033660989096931189469, 6.86619966526961928036215514509, 7.48430408852590741652075899085, 8.181548731374789339159538236500, 9.022037321098964766725752589870, 9.892456110421411880520808687134, 11.40633999796308638344719641538

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