↑

Properties

Label 2-684-4.3-c2-0-45
Degree $2$
Conductor $684$
Sign $0.791 - 0.611i$
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  + (−1.79 − 0.881i)2-s + (2.44 + 3.16i)4-s + 9.55·5-s + 12.8i·7-s + (−1.59 − 7.83i)8-s + (−17.1 − 8.42i)10-s + 1.12i·11-s + 7.90·13-s + (11.3 − 23.0i)14-s + (−4.04 + 15.4i)16-s + 24.0·17-s + 4.35i·19-s + (23.3 + 30.2i)20-s + (0.996 − 2.02i)22-s − 11.4i·23-s + ⋯
L(s)  = 1  + (−0.897 − 0.440i)2-s + (0.611 + 0.791i)4-s + 1.91·5-s + 1.83i·7-s + (−0.199 − 0.979i)8-s + (−1.71 − 0.842i)10-s + 0.102i·11-s + 0.607·13-s + (0.809 − 1.64i)14-s + (−0.252 + 0.967i)16-s + 1.41·17-s + 0.229i·19-s + (1.16 + 1.51i)20-s + (0.0452 − 0.0921i)22-s − 0.497i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.791 - 0.611i$
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: \((2,\ 684,\ (\ :1),\ 0.791 - 0.611i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.822166867\)
\(L(\frac12)\) \(\approx\) \(1.822166867\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.79 + 0.881i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 - 9.55T + 25T^{2} \)
7 \( 1 - 12.8iT - 49T^{2} \)
11 \( 1 - 1.12iT - 121T^{2} \)
13 \( 1 - 7.90T + 169T^{2} \)
17 \( 1 - 24.0T + 289T^{2} \)
23 \( 1 + 11.4iT - 529T^{2} \)
29 \( 1 + 27.7T + 841T^{2} \)
31 \( 1 + 2.59iT - 961T^{2} \)
37 \( 1 + 27.2T + 1.36e3T^{2} \)
41 \( 1 + 39.4T + 1.68e3T^{2} \)
43 \( 1 + 3.30iT - 1.84e3T^{2} \)
47 \( 1 - 20.9iT - 2.20e3T^{2} \)
53 \( 1 + 19.2T + 2.80e3T^{2} \)
59 \( 1 - 25.7iT - 3.48e3T^{2} \)
61 \( 1 - 104.T + 3.72e3T^{2} \)
67 \( 1 + 28.7iT - 4.48e3T^{2} \)
71 \( 1 + 89.0iT - 5.04e3T^{2} \)
73 \( 1 + 38.3T + 5.32e3T^{2} \)
79 \( 1 - 141. iT - 6.24e3T^{2} \)
83 \( 1 - 37.7iT - 6.88e3T^{2} \)
89 \( 1 - 71.6T + 7.92e3T^{2} \)
97 \( 1 + 30.5T + 9.40e3T^{2} \)
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

−10.08326295498082286403301762404, −9.529719584811887405702067516061, −8.891475916944476344661007681581, −8.175458148687421304449401888281, −6.71919128844181545874902451951, −5.88387461494421106724599461272, −5.32696037351911801025091518991, −3.21239537989128317118574053584, −2.26972037241970963920781546438, −1.49887375439205623987149716734, 0.957005225940777821068781130147, 1.77142853145889152230222104929, 3.39881761666797700172472697411, 5.09171128846965020162127265970, 5.87870111969080030087534726545, 6.76945225119103551545105938347, 7.44523662098059946639152074022, 8.544339019611392869368072885690, 9.544097911509089555699592278102, 10.12088453612334881159073450277

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