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Properties

Label 2-684-4.3-c2-0-73
Degree $2$
Conductor $684$
Sign $-0.665 - 0.746i$
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  + (−0.711 − 1.86i)2-s + (−2.98 + 2.66i)4-s − 4.97·5-s − 12.2i·7-s + (7.09 + 3.68i)8-s + (3.54 + 9.30i)10-s − 13.4i·11-s + 14.1·13-s + (−22.9 + 8.72i)14-s + (1.84 − 15.8i)16-s + 5.89·17-s − 4.35i·19-s + (14.8 − 13.2i)20-s + (−25.1 + 9.57i)22-s + 0.906i·23-s + ⋯
L(s)  = 1  + (−0.355 − 0.934i)2-s + (−0.746 + 0.665i)4-s − 0.995·5-s − 1.75i·7-s + (0.887 + 0.461i)8-s + (0.354 + 0.930i)10-s − 1.22i·11-s + 1.09·13-s + (−1.63 + 0.623i)14-s + (0.115 − 0.993i)16-s + 0.346·17-s − 0.229i·19-s + (0.743 − 0.662i)20-s + (−1.14 + 0.435i)22-s + 0.0394i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.665 - 0.746i$
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: \((2,\ 684,\ (\ :1),\ -0.665 - 0.746i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6372765134\)
\(L(\frac12)\) \(\approx\) \(0.6372765134\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.711 + 1.86i)T \)
3 \( 1 \)
19 \( 1 + 4.35iT \)
good5 \( 1 + 4.97T + 25T^{2} \)
7 \( 1 + 12.2iT - 49T^{2} \)
11 \( 1 + 13.4iT - 121T^{2} \)
13 \( 1 - 14.1T + 169T^{2} \)
17 \( 1 - 5.89T + 289T^{2} \)
23 \( 1 - 0.906iT - 529T^{2} \)
29 \( 1 - 10.3T + 841T^{2} \)
31 \( 1 + 43.2iT - 961T^{2} \)
37 \( 1 + 1.61T + 1.36e3T^{2} \)
41 \( 1 + 69.3T + 1.68e3T^{2} \)
43 \( 1 - 32.0iT - 1.84e3T^{2} \)
47 \( 1 - 38.9iT - 2.20e3T^{2} \)
53 \( 1 - 8.31T + 2.80e3T^{2} \)
59 \( 1 - 20.9iT - 3.48e3T^{2} \)
61 \( 1 + 118.T + 3.72e3T^{2} \)
67 \( 1 + 57.4iT - 4.48e3T^{2} \)
71 \( 1 - 11.3iT - 5.04e3T^{2} \)
73 \( 1 + 23.5T + 5.32e3T^{2} \)
79 \( 1 - 0.286iT - 6.24e3T^{2} \)
83 \( 1 + 24.9iT - 6.88e3T^{2} \)
89 \( 1 - 43.7T + 7.92e3T^{2} \)
97 \( 1 - 115.T + 9.40e3T^{2} \)
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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

−9.986573464650771195950134376365, −8.854548751408863094897181263453, −7.994563219284779894445945750763, −7.52918129781342131689551920519, −6.24236159710112509452645887354, −4.62691892826933773137912002358, −3.78584347269441346206524519297, −3.26262019375012226970331676750, −1.21769723287882998510702962190, −0.30224066522737491407661488163, 1.73479722321722610310385834901, 3.44728250232358340823856583552, 4.68315024039399351182036432756, 5.52105548052745348666244375902, 6.47132975572537259171620526965, 7.37715307684658278180825614255, 8.389600572724155250076198836338, 8.731119608155811650237851152338, 9.725517618539229733058180343367, 10.65189337941285363334811771264

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