↑

Properties

Label 2-684-4.3-c2-0-1
Degree $2$
Conductor $684$
Sign $-0.981 - 0.193i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.54 + 1.27i)2-s + (0.773 − 3.92i)4-s − 7.21·5-s − 4.48i·7-s + (3.78 + 7.04i)8-s + (11.1 − 9.15i)10-s − 15.3i·11-s + 5.21·13-s + (5.69 + 6.92i)14-s + (−14.8 − 6.07i)16-s + 4.90·17-s + 4.35i·19-s + (−5.57 + 28.2i)20-s + (19.5 + 23.7i)22-s + 0.712i·23-s + ⋯
L(s)  = 1  + (−0.772 + 0.635i)2-s + (0.193 − 0.981i)4-s − 1.44·5-s − 0.640i·7-s + (0.473 + 0.880i)8-s + (1.11 − 0.915i)10-s − 1.39i·11-s + 0.401·13-s + (0.406 + 0.494i)14-s + (−0.925 − 0.379i)16-s + 0.288·17-s + 0.229i·19-s + (−0.278 + 1.41i)20-s + (0.887 + 1.07i)22-s + 0.0309i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.981 - 0.193i$
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: \((2,\ 684,\ (\ :1),\ -0.981 - 0.193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.02597415542\)
\(L(\frac12)\) \(\approx\) \(0.02597415542\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.54 - 1.27i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 7.21T + 25T^{2} \)
7 \( 1 + 4.48iT - 49T^{2} \)
11 \( 1 + 15.3iT - 121T^{2} \)
13 \( 1 - 5.21T + 169T^{2} \)
17 \( 1 - 4.90T + 289T^{2} \)
23 \( 1 - 0.712iT - 529T^{2} \)
29 \( 1 - 20.1T + 841T^{2} \)
31 \( 1 + 24.8iT - 961T^{2} \)
37 \( 1 + 58.0T + 1.36e3T^{2} \)
41 \( 1 + 61.7T + 1.68e3T^{2} \)
43 \( 1 - 83.9iT - 1.84e3T^{2} \)
47 \( 1 + 30.9iT - 2.20e3T^{2} \)
53 \( 1 + 75.5T + 2.80e3T^{2} \)
59 \( 1 + 0.317iT - 3.48e3T^{2} \)
61 \( 1 - 87.9T + 3.72e3T^{2} \)
67 \( 1 - 51.4iT - 4.48e3T^{2} \)
71 \( 1 - 40.2iT - 5.04e3T^{2} \)
73 \( 1 + 113.T + 5.32e3T^{2} \)
79 \( 1 + 28.4iT - 6.24e3T^{2} \)
83 \( 1 - 105. iT - 6.88e3T^{2} \)
89 \( 1 + 103.T + 7.92e3T^{2} \)
97 \( 1 + 151.T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.70151604634272796998244689175, −9.787537126191793163863954536129, −8.444325091551769950380699292979, −8.301901332057032784563300725336, −7.32384871824445182626869070731, −6.51876768340825476506314543813, −5.43668310753228919005862957031, −4.20998197184576044601920340532, −3.21672035477927233604937953110, −1.10168958786689728599977915353, 0.01447486956005293847507324670, 1.72238759238022374287444282152, 3.07860714165843585974700002398, 4.01423918367397900643924539922, 5.01750713358229741320933040840, 6.79769197461693773677125496096, 7.38270160379255193613668898321, 8.351748578092812924452772078741, 8.865022954915167764655875478667, 9.973032980920822383946306427648

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