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Properties

Label 1-60e2-3600.1067-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.705 + 0.708i$
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 + (0.994 − 0.104i)11-s + (0.104 − 0.994i)13-s + (0.951 − 0.309i)17-s + (0.951 − 0.309i)19-s + (−0.406 + 0.913i)23-s + (0.743 − 0.669i)29-s + (−0.669 + 0.743i)31-s + (0.809 + 0.587i)37-s + (−0.104 + 0.994i)41-s + (0.5 − 0.866i)43-s + (−0.743 + 0.669i)47-s + (0.5 + 0.866i)49-s + (−0.309 + 0.951i)53-s + (0.994 + 0.104i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.994 − 0.104i)11-s + (0.104 − 0.994i)13-s + (0.951 − 0.309i)17-s + (0.951 − 0.309i)19-s + (−0.406 + 0.913i)23-s + (0.743 − 0.669i)29-s + (−0.669 + 0.743i)31-s + (0.809 + 0.587i)37-s + (−0.104 + 0.994i)41-s + (0.5 − 0.866i)43-s + (−0.743 + 0.669i)47-s + (0.5 + 0.866i)49-s + (−0.309 + 0.951i)53-s + (0.994 + 0.104i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.705 + 0.708i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.705 + 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.871030998 + 0.7769200310i\)
\(L(\frac12)\) \(\approx\) \(1.871030998 + 0.7769200310i\)
\(L(1)\) \(\approx\) \(1.085881100 - 0.03274605126i\)
\(L(1)\) \(\approx\) \(1.085881100 - 0.03274605126i\)

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

−18.43955259934303956664778695257, −17.9111125351000822680968035049, −16.77859403034496983461017245988, −16.43327852630045687064149659874, −15.920752862379905455774306935358, −14.76572440723615521948422016958, −14.45687709915560751820778392466, −13.663178820632810465909881545345, −12.78756203361291928251097321582, −12.111795909052011054245733694632, −11.73356492131562193191298326294, −10.74188991313422543740205729059, −9.85500101334099815637658213842, −9.366817763158810672428555646271, −8.748629705068806163633964450876, −7.81662860554851437960449591574, −6.9598298768384823227287398834, −6.30748811247199273373972413273, −5.73253584987019494998179338372, −4.724366375249273683022318073842, −3.82413387784547019155801202872, −3.2792609322404542580118964061, −2.24036401591632003222166672468, −1.417151991229203428765213994865, −0.38413801625542359098768114705, 0.83627030288925970172323519045, 1.28894236051822959744368520254, 2.78107517836093549711216510331, 3.3105236603215957046760576560, 4.004234165636809290214494956733, 5.01658498626213554220355285256, 5.853989138408245046510349096977, 6.44093438578732429121982916791, 7.380796124175789632400444796110, 7.826755784755320822227010759593, 8.89218385353679465395984140556, 9.64594458198377662026594357264, 10.04333935063066429339169283223, 10.92207547929272805894516506820, 11.794376218642292410700310502793, 12.28261191236723300700805165494, 13.20660332789423132105678998226, 13.70912495732301907796216868524, 14.42080384825560190927130116481, 15.20038525325574470587071058714, 16.05589893364892240790757732699, 16.39254386393908781417830580860, 17.31892126953567040675289451242, 17.78662752375715722538321716156, 18.67050721744291543866856870999

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