↑

Properties

Label 1-20e2-400.19-r1-0-0
Degree $1$
Conductor $400$
Sign $0.695 - 0.718i$
Analytic cond. $42.9859$
Root an. cond. $42.9859$
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  + (0.951 + 0.309i)3-s − 7-s + (0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + (−0.587 + 0.809i)13-s + (−0.309 − 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.951 − 0.309i)21-s + (0.809 − 0.587i)23-s + (0.587 + 0.809i)27-s + (0.951 + 0.309i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (0.587 − 0.809i)37-s + (−0.809 + 0.587i)39-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s − 7-s + (0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + (−0.587 + 0.809i)13-s + (−0.309 − 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.951 − 0.309i)21-s + (0.809 − 0.587i)23-s + (0.587 + 0.809i)27-s + (0.951 + 0.309i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (0.587 − 0.809i)37-s + (−0.809 + 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(42.9859\)
Root analytic conductor: \(42.9859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (1:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.012775065 - 0.8523001352i\)
\(L(\frac12)\) \(\approx\) \(2.012775065 - 0.8523001352i\)
\(L(1)\) \(\approx\) \(1.310345498 - 0.05959132413i\)
\(L(1)\) \(\approx\) \(1.310345498 - 0.05959132413i\)

Euler product

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

−24.45037079250615322764614489395, −23.40227995407263790856313612787, −22.625126708643012461277330739443, −21.58227700913189005622022420676, −20.648023409230509665927492436, −19.74998939911435483435294920670, −19.36047893268405570485143830782, −18.19983518983945003966138795272, −17.481183640163279463282910624322, −16.1128910237282606210318930585, −15.36302487905900687259377953344, −14.63892164911896134556949517226, −13.438144193615184300906269140978, −12.846497718189364277409098061558, −12.11418427494737843682191582856, −10.413609315578160554584968144482, −9.80879646495990505346848026791, −8.85443746108263474971184445867, −7.72958202806629438312571838003, −7.07409665003091674456252239480, −5.87455853813626027588888922717, −4.51942314558342973875764886029, −3.25771261161873318126811336665, −2.57650140521296658891647326651, −1.138959177085353755187636655769, 0.57670887346634361328978160007, 2.43917890390805785506546588679, 3.0687715577722774055351565632, 4.239503064406695818981535655322, 5.361867350030728726439494128399, 6.77797478110741379185052662848, 7.55033694944991455635432134051, 8.82992197452902011747276851461, 9.410467374973932059000345515445, 10.31466677248660124861003591304, 11.44363396782783252557402971880, 12.68524174859136804725030946630, 13.5441669774983141175930114073, 14.16136962149338383995270145315, 15.30253899928449205561278505684, 16.118903966712148167638458316772, 16.65364089032828174369958261171, 18.2635142457167542948367770800, 18.942007596073528548401582840726, 19.72874803141430917739628308665, 20.48539228553854070794185546023, 21.49551502358375356159956810790, 22.11179815082015297746954928576, 23.14534059096695649691219764096, 24.307059872961918171809906184668

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