↑

Properties

Label 1-60e2-3600.2309-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.563 - 0.826i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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  + (−0.5 + 0.866i)7-s + (−0.207 + 0.978i)11-s + (−0.207 − 0.978i)13-s + (−0.809 − 0.587i)17-s + (−0.587 + 0.809i)19-s + (−0.669 + 0.743i)23-s + (0.994 − 0.104i)29-s + (−0.104 + 0.994i)31-s + (−0.951 − 0.309i)37-s + (−0.978 + 0.207i)41-s + (0.866 + 0.5i)43-s + (−0.104 − 0.994i)47-s + (−0.5 − 0.866i)49-s + (−0.587 − 0.809i)53-s + (0.207 + 0.978i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (−0.207 + 0.978i)11-s + (−0.207 − 0.978i)13-s + (−0.809 − 0.587i)17-s + (−0.587 + 0.809i)19-s + (−0.669 + 0.743i)23-s + (0.994 − 0.104i)29-s + (−0.104 + 0.994i)31-s + (−0.951 − 0.309i)37-s + (−0.978 + 0.207i)41-s + (0.866 + 0.5i)43-s + (−0.104 − 0.994i)47-s + (−0.5 − 0.866i)49-s + (−0.587 − 0.809i)53-s + (0.207 + 0.978i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.563 - 0.826i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3452736509 - 0.1824273886i\)
\(L(\frac12)\) \(\approx\) \(0.3452736509 - 0.1824273886i\)
\(L(1)\) \(\approx\) \(0.7598570714 + 0.1549162619i\)
\(L(1)\) \(\approx\) \(0.7598570714 + 0.1549162619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (-0.207 - 0.978i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (0.994 - 0.104i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (-0.207 + 0.978i)T \)
67 \( 1 + (-0.994 - 0.104i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.406 - 0.913i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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

−18.864620538532615145392122570780, −17.8575888645579881283994503189, −17.09241141333927192904400471796, −16.70748413155527158566047116849, −15.86244321219221546300892388179, −15.38707184456426620194220398628, −14.19873343923387448337852158537, −13.91236175359545393007584393675, −13.16050702338192543790334473664, −12.49653322398727734087922225554, −11.59701114249681333389912870067, −10.84614361349280412971592272136, −10.42548619108275503664021227366, −9.48023409101159569607868480171, −8.7785607201136249406908756965, −8.12844599250256603736307353759, −7.17316455264236546528488137378, −6.48001983883737966893325666366, −6.050060325589188346703393155992, −4.73903833042239589494503918692, −4.26318013697427803734832882355, −3.42184784219080179236450329613, −2.54488109888344356994991324069, −1.6484938387528800402024756531, −0.49051238965989770309083077657, 0.09984303763642363412058950797, 1.48767864912592050169678613283, 2.32082985355920518070211617479, 2.99547785400222084338028527116, 3.90917307680751800596363214885, 4.886308600617036022069344718309, 5.46760416748177138708185242896, 6.31000360099508218930711709269, 7.02212660941044756105726311347, 7.839884078123512774574551086324, 8.61413629758419175744205094415, 9.258263636574723722718165316832, 10.22185936308840205064727181161, 10.400769677449279783654912657509, 11.77908540682692607726192357651, 12.09079440608791979386270193493, 12.88049795299444658739713105190, 13.411259753459014228052485030686, 14.44523411256522168426615072953, 15.02730545102960251644040976687, 15.792053920056282426348734831844, 16.03979923611156416338569868514, 17.22091103753141626525376960900, 17.860092591551616712759100351790, 18.2098672934351545519843428003

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