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Properties

Label 1-60e2-3600.1771-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.00174 + 0.999i$
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 + (0.994 + 0.104i)11-s + (−0.994 + 0.104i)13-s + (0.309 − 0.951i)17-s + (0.951 + 0.309i)19-s + (0.913 − 0.406i)23-s + (−0.743 − 0.669i)29-s + (−0.669 − 0.743i)31-s + (−0.587 − 0.809i)37-s + (0.104 + 0.994i)41-s + (−0.866 + 0.5i)43-s + (−0.669 + 0.743i)47-s + (−0.5 + 0.866i)49-s + (−0.951 + 0.309i)53-s + (−0.994 + 0.104i)59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.994 + 0.104i)11-s + (−0.994 + 0.104i)13-s + (0.309 − 0.951i)17-s + (0.951 + 0.309i)19-s + (0.913 − 0.406i)23-s + (−0.743 − 0.669i)29-s + (−0.669 − 0.743i)31-s + (−0.587 − 0.809i)37-s + (0.104 + 0.994i)41-s + (−0.866 + 0.5i)43-s + (−0.669 + 0.743i)47-s + (−0.5 + 0.866i)49-s + (−0.951 + 0.309i)53-s + (−0.994 + 0.104i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.00174 + 0.999i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.00174 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5364959461 + 0.5355604002i\)
\(L(\frac12)\) \(\approx\) \(0.5364959461 + 0.5355604002i\)
\(L(1)\) \(\approx\) \(0.9250046224 - 0.1082784576i\)
\(L(1)\) \(\approx\) \(0.9250046224 - 0.1082784576i\)

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 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.994 + 0.104i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (-0.743 - 0.669i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (-0.994 + 0.104i)T \)
61 \( 1 + (0.994 + 0.104i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−18.45918338980309700912869176171, −17.57418243530752884974359890594, −16.984552816764955037859657853050, −16.37774584879180893047330054780, −15.49099621652850637771221749196, −14.93198610805942024345323412189, −14.37244162744369785197863186833, −13.47672881652663139566646077417, −12.67788989139535512922374675945, −12.143169666865064618993469195162, −11.5593066764661403606896257311, −10.65845819968689928719752658154, −9.8252408351319145343460349187, −9.17981725889630963531263678786, −8.71158553118179886319374727758, −7.67405507217469943668725073657, −6.92395710055212207325697226581, −6.29840147035139086882428444471, −5.31517590883099308833439700911, −4.94226577865050605744090925561, −3.46778952133554706915038486148, −3.335575250245103152245023208103, −2.07868671418911826183673652718, −1.39093432584260807766618641794, −0.138984936595756885222014636, 0.76724278366142083878420614178, 1.59212701176777055512776412869, 2.74954065124603369163214594214, 3.43466516009337863183919757148, 4.27188138182830515119254111263, 4.96500463556659644715789937549, 5.884219019254125393938294433437, 6.78840121901966669710778823031, 7.294720231873279572853712281263, 7.89974911137673008289079133906, 9.14784502552163150958077720517, 9.59411656586023675008046240820, 10.10131789018496558731000270390, 11.2431824556906183054170713028, 11.596831521891222363608451585520, 12.56842267446317115232488852412, 13.092730330474747811973510528922, 14.060730638056141150887752442622, 14.39206177855542128955580179251, 15.18232427253885410887400998315, 16.17198920083248781235424386770, 16.66655274754685850290723911993, 17.1764312453605627647760158649, 17.935160346619855875364361712453, 18.84982901284363275059232823805

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