↑

Properties

Label 1-40e2-1600.1429-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.191 + 0.981i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (0.649 + 0.760i)3-s + (−0.707 − 0.707i)7-s + (−0.156 + 0.987i)9-s + (−0.522 − 0.852i)11-s + (0.852 + 0.522i)13-s + (0.951 + 0.309i)17-s + (−0.996 + 0.0784i)19-s + (0.0784 − 0.996i)21-s + (−0.987 + 0.156i)23-s + (−0.852 + 0.522i)27-s + (0.649 + 0.760i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.233 + 0.972i)37-s + (0.156 + 0.987i)39-s + ⋯
L(s)  = 1  + (0.649 + 0.760i)3-s + (−0.707 − 0.707i)7-s + (−0.156 + 0.987i)9-s + (−0.522 − 0.852i)11-s + (0.852 + 0.522i)13-s + (0.951 + 0.309i)17-s + (−0.996 + 0.0784i)19-s + (0.0784 − 0.996i)21-s + (−0.987 + 0.156i)23-s + (−0.852 + 0.522i)27-s + (0.649 + 0.760i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.233 + 0.972i)37-s + (0.156 + 0.987i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.191 + 0.981i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.191 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9334138489 + 1.132824992i\)
\(L(\frac12)\) \(\approx\) \(0.9334138489 + 1.132824992i\)
\(L(1)\) \(\approx\) \(1.085064072 + 0.3617709293i\)
\(L(1)\) \(\approx\) \(1.085064072 + 0.3617709293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.649 + 0.760i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.522 - 0.852i)T \)
13 \( 1 + (0.852 + 0.522i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.996 + 0.0784i)T \)
23 \( 1 + (-0.987 + 0.156i)T \)
29 \( 1 + (0.649 + 0.760i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.233 + 0.972i)T \)
41 \( 1 + (0.156 - 0.987i)T \)
43 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.0784 - 0.996i)T \)
59 \( 1 + (0.233 + 0.972i)T \)
61 \( 1 + (0.972 + 0.233i)T \)
67 \( 1 + (-0.0784 - 0.996i)T \)
71 \( 1 + (-0.453 + 0.891i)T \)
73 \( 1 + (0.987 - 0.156i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.996 - 0.0784i)T \)
89 \( 1 + (0.156 + 0.987i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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

−20.25295828036549198587780449265, −19.40317765011658309100161556698, −18.81458177937381131604907401282, −18.15170863015608121564688977355, −17.59156330686119464702788292259, −16.42684075616369436799937755487, −15.63250486984798861769735458848, −15.03359970452217186621309000014, −14.255382527749889876510248160561, −13.32955643509972178765320201351, −12.73024905941062204955897383573, −12.26224311347493599704404198830, −11.298150693006772802838741015207, −10.08618284955479783239497586718, −9.57998317933486241672439276363, −8.574433895004318983400702340419, −7.99028133808232476028104796997, −7.1919879716365050609662599588, −6.16951903773687553005630722275, −5.75255967318916445882332049261, −4.34757757827916537227072048793, −3.405104296630422315310023345560, −2.54594285779813342689237532323, −1.89422950912196156340004233755, −0.51170268913504311451464523802, 1.187883771571008722847458282915, 2.4406096184207715582856776272, 3.48782869375649080598282981917, 3.79195025054735898155805574725, 4.8709803445552351566691011297, 5.87798338131294364100730484759, 6.6641264353067249440667292579, 7.80658657350048173026998729293, 8.438741398878424890126056752050, 9.16369664324786456137179179900, 10.20225823220667496481405688557, 10.50184469330971338488869078188, 11.366261962538290256686384200918, 12.559063156334678284466061114129, 13.33293756788393888615082409358, 14.00828828377775733166949822266, 14.53535040855673295767208205307, 15.632295237096021860427330690294, 16.30453491503038868244022185650, 16.51652487461800034394626490071, 17.646051521143456116710781279114, 18.741411888584333153236828529420, 19.28412585644836986460498087282, 19.89897391455970096148852235074, 20.83669431373641787038182244601

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