↑

Properties

Label 2-820-820.139-c0-0-1
Degree $2$
Conductor $820$
Sign $0.995 + 0.0984i$
Analytic cond. $0.409233$
Root an. cond. $0.639713$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + 1.61·3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.500 + 1.53i)6-s + (−0.618 − 1.90i)7-s + (0.809 − 0.587i)8-s + 1.61·9-s + (0.809 − 0.587i)10-s + (−1.30 − 0.951i)12-s + 1.99·14-s + (−1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.500 + 1.53i)18-s + (0.309 + 0.951i)20-s + (−1.00 − 3.07i)21-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + 1.61·3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.500 + 1.53i)6-s + (−0.618 − 1.90i)7-s + (0.809 − 0.587i)8-s + 1.61·9-s + (0.809 − 0.587i)10-s + (−1.30 − 0.951i)12-s + 1.99·14-s + (−1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.500 + 1.53i)18-s + (0.309 + 0.951i)20-s + (−1.00 − 3.07i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.995 + 0.0984i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ 0.995 + 0.0984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083442235\)
\(L(\frac12)\) \(\approx\) \(1.083442235\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 - 1.61T + T^{2} \)
7 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.61T + T^{2} \)
89 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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

−10.08039803424790952589906146721, −9.307944094663901694399844243433, −8.603544865556118154619867719309, −7.72034769269069050064926221517, −7.42885218571597375905986078273, −6.53281417087198642667966061155, −4.79399823559977154380091261867, −4.00785381160780615597766451765, −3.34263618843484580451267197172, −1.16562337321160233449199782908, 2.25509758017297986770634932613, 2.77680952240398615332876386539, 3.52365474074702603368951110001, 4.61802746560762849090724436068, 6.19557549911782811728553112811, 7.48271776207967215299945373255, 8.375750861455061571822683626075, 8.685346305276017232758761057505, 9.532992246088316991938534810972, 10.18194207779806716219968210900

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