Properties

Label 1-2736-2736.2189-r1-0-0
Degree $1$
Conductor $2736$
Sign $0.863 + 0.504i$
Analytic cond. $294.024$
Root an. cond. $294.024$
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  + (−0.642 + 0.766i)5-s + (0.5 − 0.866i)7-s − i·11-s + (0.642 + 0.766i)13-s + (−0.766 − 0.642i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s + 31-s + (0.342 + 0.939i)35-s − i·37-s + (0.173 − 0.984i)41-s + (0.342 + 0.939i)43-s + (0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)5-s + (0.5 − 0.866i)7-s − i·11-s + (0.642 + 0.766i)13-s + (−0.766 − 0.642i)17-s + (−0.939 − 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s + 31-s + (0.342 + 0.939i)35-s − i·37-s + (0.173 − 0.984i)41-s + (0.342 + 0.939i)43-s + (0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.863 + 0.504i$
Analytic conductor: \(294.024\)
Root analytic conductor: \(294.024\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (1:\ ),\ 0.863 + 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726443329 + 0.4669134255i\)
\(L(\frac12)\) \(\approx\) \(1.726443329 + 0.4669134255i\)
\(L(1)\) \(\approx\) \(0.9852034643 + 0.1087162148i\)
\(L(1)\) \(\approx\) \(0.9852034643 + 0.1087162148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−19.06585778855809308588898119095, −18.38615221047441752395753173235, −17.579979946580195508788407652639, −16.94823252977144081720156593614, −16.038364805223237173667973127737, −15.48853670404097586085267059206, −15.10661006007875331567500598060, −13.91513317073797760178778608501, −13.33819554259255659823976217997, −12.57162793832079495064249856266, −11.798685614179698272006518938554, −11.33405021498374476736799395479, −10.53779948531621311257875210380, −9.499702044592277089970709717090, −8.620031694897562135586105019355, −8.28522137497263179906883131182, −7.702868731543829705433422555283, −6.2903705797045027791868764243, −5.82406823664026613644208932258, −4.98247140377399101222343365431, −4.15026513949900875305811345026, −3.38774363686350537685700099409, −2.38885359213444967955680134371, −1.37448816104598531183674872067, −0.498552176929353942653988386336, 0.553074123580429275268654276109, 1.71871758064986657384234360621, 2.49401970540985385282019209914, 3.62767642247040414955472137043, 4.26351631953873134485216537916, 4.801280697515885554389051907261, 6.1399344276568768844367377930, 6.87643890160344370866493476628, 7.390103069542501071992260792798, 8.088106497977083238847842477377, 9.03750678328068677672510532698, 9.86699044744714269355065611154, 10.76144341833585642811185992442, 11.093436771400585924493326292951, 11.94559645124413356028347927704, 12.621932174867119496668450995887, 13.76856132978323591299329332517, 14.117380576041085785856476618013, 14.84287759626385716753048137650, 15.6898068733475392222791873460, 16.167276838598581694801000231945, 17.099132344489408631479620276010, 18.007506106116764774250182541955, 18.16649820441412821750172669651, 19.26765462578974014606706087143

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