Properties

Label 2-567-7.4-c1-0-15
Degree $2$
Conductor $567$
Sign $0.956 - 0.290i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.247 + 0.429i)2-s + (0.877 − 1.51i)4-s + (1.84 + 3.19i)5-s + (1.68 − 2.04i)7-s + 1.86·8-s + (−0.915 + 1.58i)10-s + (0.446 − 0.772i)11-s − 1.19·13-s + (1.29 + 0.216i)14-s + (−1.29 − 2.23i)16-s + (−0.124 + 0.216i)17-s + (1.40 + 2.43i)19-s + 6.47·20-s + 0.442·22-s + (−1.23 − 2.14i)23-s + ⋯
L(s)  = 1  + (0.175 + 0.303i)2-s + (0.438 − 0.759i)4-s + (0.825 + 1.43i)5-s + (0.636 − 0.771i)7-s + 0.658·8-s + (−0.289 + 0.501i)10-s + (0.134 − 0.233i)11-s − 0.331·13-s + (0.345 + 0.0578i)14-s + (−0.323 − 0.559i)16-s + (−0.0303 + 0.0525i)17-s + (0.322 + 0.557i)19-s + 1.44·20-s + 0.0943·22-s + (−0.258 − 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.956 - 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14590 + 0.318233i\)
\(L(\frac12)\) \(\approx\) \(2.14590 + 0.318233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-1.68 + 2.04i)T \)
good2 \( 1 + (-0.247 - 0.429i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.84 - 3.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.446 + 0.772i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.19T + 13T^{2} \)
17 \( 1 + (0.124 - 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.40 - 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.14T + 29T^{2} \)
31 \( 1 + (1.79 - 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.78T + 41T^{2} \)
43 \( 1 - 9.97T + 43T^{2} \)
47 \( 1 + (-5.08 - 8.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.94 - 8.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.906 - 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.40 + 9.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.514 - 0.891i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (0.915 - 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.899 - 1.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + (1.20 + 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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

−10.87810189219608019783881362757, −10.12764305275506122048412111670, −9.309256767419012659863718567117, −7.66250502125030668063762218026, −7.15368210802432678637470835064, −6.19898055204714774311799313277, −5.54844835501179322555718763912, −4.20802142504761672680008702949, −2.74246329292561161040652167153, −1.58134035456493381990220559849, 1.58706660194639455794579954313, 2.52363969580204646353139037546, 4.14484042475261234181761442525, 5.07505186410702096750933535820, 5.86797822489577958818047326051, 7.26721204101600807103288791006, 8.177984890292734661376935446608, 8.985314971701460495757305647820, 9.615656902855018672546922713636, 10.89547725973241553117055480119

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