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Properties

Label 1-60e2-3600.13-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.185 - 0.982i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.185 - 0.982i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.185 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1199389177 - 0.1447239148i\)
\(L(\frac12)\) \(\approx\) \(0.1199389177 - 0.1447239148i\)
\(L(1)\) \(\approx\) \(0.7878736674 + 0.1027059839i\)
\(L(1)\) \(\approx\) \(0.7878736674 + 0.1027059839i\)

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

−18.57329115708739321796016845087, −18.28252689988264716576709136970, −17.32710176044373948796568727789, −16.48806236654291954261692312053, −15.92660370535122525825521153042, −15.52238655663905765938889390034, −14.61327356705574304902225475029, −13.59539265719539740781186886201, −13.35399291495310537136647855821, −12.62008972111288568235156409892, −11.529031448760151710847244760466, −11.260052547772189431555872201881, −10.37530624721895286960701934811, −9.37429563364031055046224789209, −9.036204698013541167733453608892, −8.24479352338328762625686441447, −7.34934445115023577743283274692, −6.48386177488944727924963391495, −5.93692140310538397019244045512, −5.26723803305113802990872335118, −4.11087354884473978485606446989, −3.4962328211197348590416683912, −2.652682647946589778691403192304, −1.87886769639425575895325432174, −0.56917904310210182076830105887, 0.04333617818289596991233255670, 1.33038085466688906788650092444, 2.03736568587796791742932147645, 3.13524034112308351770839803077, 3.86912378811835956059739131149, 4.46075060970903576080122459251, 5.51207818663207572968214276018, 6.48112253947031261504646965276, 6.65436675173412285361119657727, 7.866134039826303571965574813044, 8.32161286022161472706328942150, 9.339834365493569810340388116431, 9.95400713554682169525577671989, 10.64274805901525529339470175775, 11.16918297621330165845588166265, 12.51314640976001500244620638347, 12.603701798355747549311027829, 13.406284188831299760044608102580, 14.193172760671453137904822687, 14.93698184452728448458597557228, 15.654998079696112181080533595672, 16.24545508315393662965791440922, 16.935828819196216571039613450, 17.64552872788405120067686427407, 18.33944218327548773752103280974

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