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Properties

Label 1-60e2-3600.2011-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.859 + 0.510i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.743 − 0.669i)11-s + (−0.743 − 0.669i)13-s + (−0.809 + 0.587i)17-s + (−0.587 − 0.809i)19-s + (−0.978 + 0.207i)23-s + (−0.406 − 0.913i)29-s + (−0.913 − 0.406i)31-s + (−0.951 + 0.309i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + (−0.913 + 0.406i)47-s + (−0.5 − 0.866i)49-s + (0.587 − 0.809i)53-s + (−0.743 − 0.669i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.743 − 0.669i)11-s + (−0.743 − 0.669i)13-s + (−0.809 + 0.587i)17-s + (−0.587 − 0.809i)19-s + (−0.978 + 0.207i)23-s + (−0.406 − 0.913i)29-s + (−0.913 − 0.406i)31-s + (−0.951 + 0.309i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + (−0.913 + 0.406i)47-s + (−0.5 − 0.866i)49-s + (0.587 − 0.809i)53-s + (−0.743 − 0.669i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.859 + 0.510i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.859 + 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7392253454 + 0.2029227109i\)
\(L(\frac12)\) \(\approx\) \(0.7392253454 + 0.2029227109i\)
\(L(1)\) \(\approx\) \(0.7808238373 + 0.001054727116i\)
\(L(1)\) \(\approx\) \(0.7808238373 + 0.001054727116i\)

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

Imaginary part of the first few zeros on the critical line

−18.44259191635737612339311308445, −17.70088972199067768965499538033, −17.01933464523962927568056599536, −16.49465075990656397465462621133, −15.86215310853427068658958444222, −14.88117009476877792202326597498, −14.30150687339438292497354463522, −13.76415732540579465211533400316, −12.8656586317554232031631407425, −12.2233304193968642004774064484, −11.65426045119754341943432498596, −10.54613596999135543794196110497, −10.23877082010276391290661242912, −9.23095406318800871717317731249, −8.877977561041557848797708865959, −7.62243402220394658367794551171, −7.06713452283833800091888813631, −6.58240796736866581577897591064, −5.60876852291293094605792907572, −4.578963431451245071908863952588, −4.07545734492726317010420990590, −3.32838898373954985791992863666, −2.118929407904061419759335688831, −1.57790111666998514256485283997, −0.24627762529477975560497956249, 0.37669060412983191298622490868, 1.73931351265943562130163713762, 2.43878671255288064806122229668, 3.28919820024595739140227283778, 4.07512224247869232288861342696, 4.985523126896779403814834864722, 5.872540347107827009086011996916, 6.33255173168905173398132477297, 7.160192995689601073267963312528, 8.18824012863302651841923199675, 8.691266750320560610155725264807, 9.4939118087987486330365000394, 10.04331555644643213549027591168, 11.09349245536403012674123100338, 11.57121637062962915334326220899, 12.42356430271192695459419625845, 12.99498616415334142750492415772, 13.64863759818716113833437319544, 14.65950095029067427590417970519, 15.097614365863527259919169889084, 15.80661867357854003616960760887, 16.47507033327212560959167945497, 17.35571929709393182232776905393, 17.7407480617859821801685820907, 18.69151501062431172484581056074

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