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Properties

Label 1-1368-1368.587-r0-0-0
Degree $1$
Conductor $1368$
Sign $-0.327 + 0.944i$
Analytic cond. $6.35296$
Root an. cond. $6.35296$
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  + (0.173 + 0.984i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.766 + 0.642i)29-s + (0.5 + 0.866i)31-s + (−0.173 − 0.984i)35-s − 37-s + (0.939 + 0.342i)41-s + (−0.939 − 0.342i)43-s + (0.766 + 0.642i)47-s + 49-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)5-s − 7-s + (0.5 − 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.766 + 0.642i)29-s + (0.5 + 0.866i)31-s + (−0.173 − 0.984i)35-s − 37-s + (0.939 + 0.342i)41-s + (−0.939 − 0.342i)43-s + (0.766 + 0.642i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(6.35296\)
Root analytic conductor: \(6.35296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (0:\ ),\ -0.327 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5658875253 + 0.7947291395i\)
\(L(\frac12)\) \(\approx\) \(0.5658875253 + 0.7947291395i\)
\(L(1)\) \(\approx\) \(0.8810242000 + 0.2208384128i\)
\(L(1)\) \(\approx\) \(0.8810242000 + 0.2208384128i\)

Euler product

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

−20.544280107087841299824645147851, −19.86132202372092679177510269220, −19.28609297455026207139530878847, −18.42819616831630275731563944573, −17.30302222276671155686808943832, −16.92275768317153502158991380271, −16.12329450708163790369314127321, −15.4835694174157513690455006423, −14.39525800474977206372856395650, −13.741170507946021333620191257957, −12.80701044635870990719279640036, −12.160469841612579111276379117172, −11.76321019473045231475390023811, −10.09803802774300429244845087832, −9.808352278744073033026229300177, −9.07746536966872485428600634316, −8.10899268220038740941466934669, −7.177940043505492494951244048630, −6.36386817454769583314737610608, −5.47312899556885777130796709467, −4.5252967912478677462162664260, −3.859907572532726589693520378834, −2.58943343663533044021668567952, −1.68846936060119883514677444078, −0.39927648895623581573107741425, 1.1764447847619108113015827618, 2.61231034989662986318763533553, 3.19794315499818493165530118335, 3.94136249481707600921961870032, 5.40478414281138936298148844854, 6.07685882467176957025745739890, 6.81670175210480215589812832824, 7.61688365557883960165837232539, 8.58340089802075408583939278605, 9.62054148087073048713954765271, 10.237334680510006633179913635669, 10.82990500951710592760145822500, 12.02774352617398140493178524546, 12.449847442000980154691289889207, 13.70562924311935990134295010551, 14.072993942722730375924906026622, 14.99178531921175210947053232786, 15.74578106533517694611863707813, 16.52997988693044119943199867711, 17.31037819485063765543318219879, 18.1283187823924243825009755511, 18.927355120982826931079702576054, 19.511093651490176579345695686601, 20.03917212174014107792132359618, 21.54734607443061363495806577393

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