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Properties

Label 1-60e2-3600.1787-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.826 - 0.563i$
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  + (−0.866 − 0.5i)7-s + (0.207 − 0.978i)11-s + (0.978 − 0.207i)13-s + (−0.587 + 0.809i)17-s + (−0.587 + 0.809i)19-s + (0.743 + 0.669i)23-s + (−0.994 + 0.104i)29-s + (0.104 − 0.994i)31-s + (−0.309 + 0.951i)37-s + (−0.978 + 0.207i)41-s + (0.5 − 0.866i)43-s + (0.994 − 0.104i)47-s + (0.5 + 0.866i)49-s + (0.809 − 0.587i)53-s + (0.207 + 0.978i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.207 − 0.978i)11-s + (0.978 − 0.207i)13-s + (−0.587 + 0.809i)17-s + (−0.587 + 0.809i)19-s + (0.743 + 0.669i)23-s + (−0.994 + 0.104i)29-s + (0.104 − 0.994i)31-s + (−0.309 + 0.951i)37-s + (−0.978 + 0.207i)41-s + (0.5 − 0.866i)43-s + (0.994 − 0.104i)47-s + (0.5 + 0.866i)49-s + (0.809 − 0.587i)53-s + (0.207 + 0.978i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.826 - 0.563i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.826 - 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2314934985 - 0.7501514503i\)
\(L(\frac12)\) \(\approx\) \(0.2314934985 - 0.7501514503i\)
\(L(1)\) \(\approx\) \(0.9028844089 - 0.1074123943i\)
\(L(1)\) \(\approx\) \(0.9028844089 - 0.1074123943i\)

Euler product

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

−18.71907580358865795293236496726, −18.232411375134281160912167049793, −17.418583255589035384905519661394, −16.767631222745683121095405191240, −15.78589535250842876371484447301, −15.62577114474120741621639991455, −14.73483031493344693241342471694, −13.958859558022153105166755630, −13.069191065417132747172437540034, −12.74827176992118949188485394974, −11.89833366635456511277687450390, −11.12620515131023232789823554271, −10.48016099015557486321033535424, −9.517511724147650829565802861664, −9.0413487687964905094901506296, −8.470022266966219762122792840676, −7.1526984603183927247630757355, −6.86337032464238560059086723558, −6.0490306286556254705300623315, −5.16578589179919629181311975608, −4.385136793463622228663965565551, −3.58048802442227215287917839142, −2.66253800023287372758276553069, −2.01603120344115012501007973133, −0.853267230900450418905573135171, 0.148077597009219182693791472599, 1.04951174170927529156100872312, 1.95143153900852996249165331424, 3.15735327684446004806720696389, 3.673456669844539417149451829838, 4.29247091572770543308426005941, 5.6592193416552276139006629150, 6.016232129644056183359829028300, 6.79435447424857495364466188025, 7.597916136088754883399086359528, 8.56852095439689862687372863319, 8.92291215301151200548607550903, 9.96028138518301105771665215353, 10.58667098462337166437145981438, 11.171884825769237319654357887166, 11.968223918426882118703271207153, 12.96440579759913386299156100360, 13.35900747998743782327343934735, 13.87988677689419138697649077102, 14.99278317722366908209410173297, 15.43423457467298182767983487539, 16.312862382725907063990383968934, 16.87836102658137671716504585927, 17.308705358255037862064241104866, 18.55259983390166912372801381659

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