↑

Properties

Label 1-4020-4020.2159-r0-0-0
Degree $1$
Conductor $4020$
Sign $-0.895 - 0.445i$
Analytic cond. $18.6688$
Root an. cond. $18.6688$
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  + (0.415 − 0.909i)7-s + (−0.142 + 0.989i)11-s + (0.959 + 0.281i)13-s + (−0.654 + 0.755i)17-s + (−0.415 − 0.909i)19-s + (−0.841 + 0.540i)23-s − 29-s + (0.959 − 0.281i)31-s − 37-s + (0.654 − 0.755i)41-s + (−0.654 + 0.755i)43-s + (−0.841 + 0.540i)47-s + (−0.654 − 0.755i)49-s + (−0.654 − 0.755i)53-s + (−0.959 + 0.281i)59-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)7-s + (−0.142 + 0.989i)11-s + (0.959 + 0.281i)13-s + (−0.654 + 0.755i)17-s + (−0.415 − 0.909i)19-s + (−0.841 + 0.540i)23-s − 29-s + (0.959 − 0.281i)31-s − 37-s + (0.654 − 0.755i)41-s + (−0.654 + 0.755i)43-s + (−0.841 + 0.540i)47-s + (−0.654 − 0.755i)49-s + (−0.654 − 0.755i)53-s + (−0.959 + 0.281i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(18.6688\)
Root analytic conductor: \(18.6688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (0:\ ),\ -0.895 - 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09735827872 - 0.4146955125i\)
\(L(\frac12)\) \(\approx\) \(0.09735827872 - 0.4146955125i\)
\(L(1)\) \(\approx\) \(0.9037264669 - 0.07052522610i\)
\(L(1)\) \(\approx\) \(0.9037264669 - 0.07052522610i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good7 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 - T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 - T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.841 + 0.540i)T \)
53 \( 1 + (-0.654 - 0.755i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.64719257481207066678004751763, −18.21915755454901948066052574757, −17.55760709988878217495185931307, −16.53074300516473658496553306942, −16.09927810870114105261792219800, −15.38503512287011283366778724325, −14.76417728520990003732794047629, −13.84731116074040095689357029664, −13.49811946115780451993023791206, −12.48471081378447956762203847907, −11.92870198154447264704986916208, −11.1542564348219042817307098530, −10.67422257607500427617678099568, −9.72785978940680772100491560579, −8.87136726172426970447987888302, −8.37540840957321610652138101459, −7.84633930641242240667014740008, −6.649749945060936201667497325963, −6.02534734553788934080773063378, −5.460999449633894576552689314436, −4.5854123688230915874290786023, −3.68063416000171370983532120996, −2.92439917454552999297395862866, −2.07708625763416876968323952958, −1.22700658751119294698925919885, 0.113085760404067990140332097250, 1.52492507352399409857397387291, 1.91187902995624133658975007596, 3.13885956722969985939421650639, 4.10014137489063006059478385342, 4.42884290359515136612234481401, 5.37163566788327305926094502274, 6.40196626742402507641742876972, 6.862453021423215976799447671433, 7.77727960145532324668534267662, 8.29044954935409385704434369925, 9.23085383427200084285015003232, 9.885598655374981936393943555237, 10.76371048330545127086963951961, 11.114973561234360883737899571559, 11.96575738486073115784884874303, 12.88096608227564632816862367682, 13.3875344868015546517423570189, 14.019888316536397887674431227567, 14.832523261812890025817356487376, 15.474568399205505815206448462422, 16.06819164615628292585540724314, 17.00493786579217375802813187918, 17.568034441929966144340859037245, 17.93061488366543837510365826678

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