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Properties

Label 2-684-4.3-c2-0-4
Degree $2$
Conductor $684$
Sign $-0.0520 - 0.998i$
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  + (−1.99 + 0.0520i)2-s + (3.99 − 0.208i)4-s − 6.84·5-s − 12.2i·7-s + (−7.97 + 0.624i)8-s + (13.6 − 0.356i)10-s − 6.17i·11-s − 24.7·13-s + (0.637 + 24.4i)14-s + (15.9 − 1.66i)16-s − 7.41·17-s + 4.35i·19-s + (−27.3 + 1.42i)20-s + (0.321 + 12.3i)22-s − 31.3i·23-s + ⋯
L(s)  = 1  + (−0.999 + 0.0260i)2-s + (0.998 − 0.0520i)4-s − 1.36·5-s − 1.74i·7-s + (−0.996 + 0.0780i)8-s + (1.36 − 0.0356i)10-s − 0.561i·11-s − 1.90·13-s + (0.0455 + 1.74i)14-s + (0.994 − 0.103i)16-s − 0.436·17-s + 0.229i·19-s + (−1.36 + 0.0712i)20-s + (0.0146 + 0.561i)22-s − 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0520 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0520 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.0520 - 0.998i$
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: \((2,\ 684,\ (\ :1),\ -0.0520 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.08123431927\)
\(L(\frac12)\) \(\approx\) \(0.08123431927\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.0520i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 6.84T + 25T^{2} \)
7 \( 1 + 12.2iT - 49T^{2} \)
11 \( 1 + 6.17iT - 121T^{2} \)
13 \( 1 + 24.7T + 169T^{2} \)
17 \( 1 + 7.41T + 289T^{2} \)
23 \( 1 + 31.3iT - 529T^{2} \)
29 \( 1 - 25.1T + 841T^{2} \)
31 \( 1 - 47.1iT - 961T^{2} \)
37 \( 1 - 12.8T + 1.36e3T^{2} \)
41 \( 1 - 32.8T + 1.68e3T^{2} \)
43 \( 1 + 37.2iT - 1.84e3T^{2} \)
47 \( 1 + 3.78iT - 2.20e3T^{2} \)
53 \( 1 + 10.8T + 2.80e3T^{2} \)
59 \( 1 - 76.1iT - 3.48e3T^{2} \)
61 \( 1 + 37.4T + 3.72e3T^{2} \)
67 \( 1 + 7.64iT - 4.48e3T^{2} \)
71 \( 1 + 30.7iT - 5.04e3T^{2} \)
73 \( 1 - 108.T + 5.32e3T^{2} \)
79 \( 1 - 34.7iT - 6.24e3T^{2} \)
83 \( 1 - 100. iT - 6.88e3T^{2} \)
89 \( 1 + 17.2T + 7.92e3T^{2} \)
97 \( 1 + 97.1T + 9.40e3T^{2} \)
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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

−10.57860100350950736306637221282, −9.765616817962512542954111260587, −8.579385145167468397119055754808, −7.84657194252671352544225628374, −7.20954715720980057979925594897, −6.64493957309593276183618140416, −4.83316178036873438652696499115, −3.91244598247521853297548727134, −2.73872291861658548646987474289, −0.833258017207484514465960284679, 0.05522077394438019461629499665, 2.14591557301009429407752560972, 2.96927096440855682193165537863, 4.54004845834967408226854416997, 5.63539051450525260206695057014, 6.83363691690345366388660355159, 7.72521605868533555764780223792, 8.183141490619840091949933476666, 9.430564029929435844971137242753, 9.579134496293842739597607081547

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