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Properties

Label 1-60e2-3600.1541-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.246 + 0.969i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.246 + 0.969i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.246 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.101066592 + 1.416931541i\)
\(L(\frac12)\) \(\approx\) \(1.101066592 + 1.416931541i\)
\(L(1)\) \(\approx\) \(1.074830370 + 0.1645287249i\)
\(L(1)\) \(\approx\) \(1.074830370 + 0.1645287249i\)

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

−18.0865457508666382145825114485, −17.79176553783158816243967168931, −16.98108784255686459782454479541, −16.196563336918186760533920914862, −15.7292059662995506315610692266, −14.7623630964262487108511663197, −14.21039162753112397236164819933, −13.44110908163621122711393146457, −12.996814557886718761677514168457, −11.91671442285518975925964267872, −11.413781602985738767342661591714, −10.59726034222269885512123473876, −10.01468065446475422560117295335, −9.27111782138745410730402924767, −8.3245070123553214581840207186, −7.66821283347784883155800590006, −7.1104574390223238676931469684, −6.20632871969603892011707199908, −5.40608258073264280872747653965, −4.37261846421028837276042103628, −4.16004435471583363700483617654, −2.926887473382402805409272872456, −2.11588595155032436051923789934, −1.19753782009442414529344336986, −0.31284696364843492950060321411, 0.874707764579378148531085613958, 1.76984855164257776956501588361, 2.66268352887901917460150945302, 3.41062343956001011316972150958, 4.28258812013729757861719445001, 5.283629428919014905316979084303, 5.88887046499537440599925668261, 6.31357136907781229552113375509, 7.832656863156002496148407422789, 8.035703480585264467169819284974, 8.700829789712363398571242083408, 9.64835719652946946956735781767, 10.48917768772762666981707790647, 10.9544145297520627781876061563, 11.847021902265901291641252069752, 12.491080590923111705334864710170, 13.117821646234719706198900552352, 13.93344708500080521724042050556, 14.71387364874365419519643654908, 15.199217808438099185996067863247, 16.05888379901071375232549957077, 16.49899863124124379295554706807, 17.455930850084024178214544963538, 18.20843832287780135984555598435, 18.59878420847091967891030879097

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