Properties

Degree 1
Conductor $ 3^{2} \cdot 13 $
Sign $0.944 + 0.329i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)10-s + 11-s + (0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯
L(s,χ)  = 1  + 2-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)10-s + 11-s + (0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.944 + 0.329i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.944 + 0.329i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(117\)    =    \(3^{2} \cdot 13\)
\( \varepsilon \)  =  $0.944 + 0.329i$
motivic weight  =  \(0\)
character  :  $\chi_{117} (101, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 117,\ (1:\ ),\ 0.944 + 0.329i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(3.427371623 + 0.5817043652i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(3.427371623 + 0.5817043652i\)
\(L(\chi,1)\)  \(\approx\)  \(2.085212009 + 0.1959125376i\)
\(L(1,\chi)\)  \(\approx\)  \(2.085212009 + 0.1959125376i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.86751324558816619622805732818, −28.1091669852299350520874876781, −27.054045901676044893309899306195, −25.31653829448427867874529718826, −24.67903301046175882984587411639, −23.91798002473662277628610515905, −22.73578034174747180538685007636, −21.8262787928231645655151995865, −20.75120461323300862884878453179, −19.99342278800781674352021054114, −18.794477091911090818354376545508, −17.15756264638828514720260094276, −16.14817694626408163685754499179, −15.19371426597113131394629437793, −14.20609858192312180274601027842, −12.92461219227665777473090590024, −11.925209487680657597038333098078, −11.34410839686814790360148792935, −9.391251114386093759911016633557, −8.17250276521063459621330051785, −6.790258393942993835953603786816, −5.329332234815875663516046692512, −4.52885830513698511424664035012, −3.02566093943063315005621156636, −1.34090902871091111756438641467, 1.57591627724811411582934508644, 3.43107393564383522955875171215, 4.149433464257809983583978894766, 5.81686422909288709047681234687, 7.03743384546344129880305677557, 7.88521381968738081726138694103, 10.052832478792924947489882472690, 11.14978369887325133860424527141, 11.89238751788743506225807230742, 13.337516040002076291869964392526, 14.452728320136641245533930895991, 14.93245191515044830870097867592, 16.36570649450617278492491331734, 17.36531848343602351666559461961, 18.999647416923577428188243609848, 19.879178313837448319128778471227, 20.93292169739913308925594984706, 22.05810345686618617914373623597, 22.93508557306845798239564382098, 23.669843474285949364417167205770, 24.7329551419102662465337362408, 25.891063708576006300243076290684, 26.91508728218746165532010031779, 27.98046338610810628304864024596, 29.59200583753848036615768641655

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