↑

Properties

Label 2-675-27.22-c1-0-22
Degree $2$
Conductor $675$
Sign $0.793 + 0.608i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (1.26 − 1.06i)2-s + (−1.44 − 0.958i)3-s + (0.124 − 0.708i)4-s + (−2.83 + 0.318i)6-s + (0.215 + 1.22i)7-s + (1.05 + 1.82i)8-s + (1.16 + 2.76i)9-s + (3.30 + 1.20i)11-s + (−0.859 + 0.902i)12-s + (0.452 + 0.380i)13-s + (1.56 + 1.31i)14-s + (4.62 + 1.68i)16-s + (2.58 − 4.46i)17-s + (4.39 + 2.26i)18-s + (1.41 + 2.45i)19-s + ⋯
L(s)  = 1  + (0.893 − 0.749i)2-s + (−0.832 − 0.553i)3-s + (0.0624 − 0.354i)4-s + (−1.15 + 0.129i)6-s + (0.0814 + 0.461i)7-s + (0.373 + 0.646i)8-s + (0.387 + 0.921i)9-s + (0.996 + 0.362i)11-s + (−0.248 + 0.260i)12-s + (0.125 + 0.105i)13-s + (0.418 + 0.351i)14-s + (1.15 + 0.420i)16-s + (0.625 − 1.08i)17-s + (1.03 + 0.533i)18-s + (0.325 + 0.563i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90448 - 0.646029i\)
\(L(\frac12)\) \(\approx\) \(1.90448 - 0.646029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.44 + 0.958i)T \)
5 \( 1 \)
good2 \( 1 + (-1.26 + 1.06i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-0.215 - 1.22i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-3.30 - 1.20i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.452 - 0.380i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.58 + 4.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.16 + 6.59i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (7.21 - 6.05i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.196 + 1.11i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.34 - 4.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.43 - 3.71i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.03 + 0.739i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.08 - 6.14i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 8.21T + 53T^{2} \)
59 \( 1 + (-12.6 + 4.62i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.38 + 7.85i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (3.37 + 2.82i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (3.43 - 5.95i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.715 - 1.23i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.970 + 0.814i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.17 + 0.982i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (6.52 + 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.0 + 4.74i)T + (74.3 + 62.3i)T^{2} \)
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

−10.82450349693667940141917369246, −9.819025785105489249673407916571, −8.684866127090504487711372926894, −7.59922486963025365961070410929, −6.71603898299867673054445831385, −5.62712712619742578488536938056, −4.92148388554705074355187140402, −3.89333716617029403091564898284, −2.59455972129561137208811260572, −1.39723328154887594716901159329, 1.12699079973699069558959835272, 3.77642383527442693489115674708, 4.02352535274562055488565037424, 5.44739506167668797371156489663, 5.78240151975581394398984091736, 6.82556736605757021887455106902, 7.52465147692236727401250662193, 8.984653110155221647660662636082, 9.843118675573927619663924885302, 10.62839600331599750807993276319

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