Properties

Label 1-6e2-36.31-r1-0-0
Degree $1$
Conductor $36$
Sign $0.173 - 0.984i$
Analytic cond. $3.86873$
Root an. cond. $3.86873$
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)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 35-s + 37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 35-s + 37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(3.86873\)
Root analytic conductor: \(3.86873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 36,\ (1:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9884724919 - 0.8294269034i\)
\(L(\frac12)\) \(\approx\) \(0.9884724919 - 0.8294269034i\)
\(L(1)\) \(\approx\) \(0.9840438113 - 0.3581626565i\)
\(L(1)\) \(\approx\) \(0.9840438113 - 0.3581626565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−35.65226917655706862275677463829, −34.38928170962187333286368319253, −33.72331345055919738890270307697, −31.974198935909409857618717886190, −30.90301731600950543144615679691, −29.9938104202461392898022778521, −28.33729745537479217726661183375, −27.34091390948471093109871095642, −26.04910307973596078760794988895, −24.827933827863368781211750821983, −23.38844143605730591265431483799, −22.24648206434101954039202024277, −21.03484422373811868900002898182, −19.32709737023475442080979923945, −18.43431621211412759036140745844, −16.93523973838671676594902327875, −15.14478864997353383792979024847, −14.48929246529189450685183375899, −12.35293697747913674603033103728, −11.29129554297552549381223229676, −9.61675299237772893847340912350, −7.90065445055288681844313381827, −6.4139388605395573100182027213, −4.395295916916735711983073054247, −2.36426765473592387475291733061, 0.93468572592624077763867749013, 3.74336585507794542185841000015, 5.32211014726077417771585399000, 7.46800994748448765693902597055, 8.72015991681402708810778978218, 10.54161693101067141682348472702, 11.991537439616035563313976849359, 13.38189167703740512806278827453, 14.85572121580761028053359824380, 16.46973826104502616682870882267, 17.34101325848358594101711625590, 19.23557391886182381803113946995, 20.26391341587413822319702352139, 21.459884940569361026427098634069, 23.19345118088844736555863741349, 24.09834201912689038847003266004, 25.31918922348646883417840897898, 27.12182721510882414006301090239, 27.637013665948661233708244495625, 29.33647098122860963978225052425, 30.31522251705824494748280115738, 31.87449655674035908979917366793, 32.603681757899809588003641453142, 34.106456299610876433114563561036, 35.29597037808016904534306015595

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