Properties

Degree 1
Conductor $ 3^{2} \cdot 5 $
Sign $0.370 + 0.929i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s i·17-s − 19-s + (−0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + 26-s + i·28-s + (0.5 − 0.866i)29-s + ⋯
L(s,χ)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)7-s + i·8-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s i·17-s − 19-s + (−0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + 26-s + i·28-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.370 + 0.929i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.370 + 0.929i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(45\)    =    \(3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.370 + 0.929i$
motivic weight  =  \(0\)
character  :  $\chi_{45} (13, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 45,\ (1:\ ),\ 0.370 + 0.929i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.089393876 + 1.416821558i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.089393876 + 1.416821558i\)
\(L(\chi,1)\)  \(\approx\)  \(1.662104067 + 0.7152142400i\)
\(L(1,\chi)\)  \(\approx\)  \(1.662104067 + 0.7152142400i\)

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

−33.563139373540307888599398043508, −32.64549932824481790078654114306, −31.3599118117517249822078823486, −30.46814483438549901161605320880, −29.42454792684348963400871208636, −28.2476323743655423292087950918, −27.01772544332566706321391337961, −25.41581901329055039096638944902, −23.84360188683474264936280512574, −23.47909827477132417934059145050, −21.643856442365695998624218737538, −21.03974624263197048446988086363, −19.67249252540702479960660775924, −18.43785434689263770344974137147, −16.69247618605106810511861226911, −15.20302890355786756304450356509, −14.01542558368522215548520337583, −12.96639624232198349239188900447, −11.3261798640330158092720241250, −10.54877009726074926044277444140, −8.456015699452004513454125940380, −6.5293187531311927236861314421, −4.98797822013136312246649457280, −3.52118562865846980276665911532, −1.49571308205630303629853667443, 2.430404657581143479635392788174, 4.41542201283699230114209826680, 5.671260819630638130284355295785, 7.33113221226391634339523192714, 8.637184166945910819260063887699, 10.87606403167583598028331885299, 12.19459817744109971385172785361, 13.425728737179964487181909763810, 14.82456003075162729769204335935, 15.66433874171719306735619303876, 17.22403862792126177208631829324, 18.38262166361494488055232062814, 20.48414765842990332426277294512, 21.184197901919040255671317947, 22.645703368796494131975320336025, 23.564101765526483804325748135303, 24.857456588195913533412338521291, 25.664790105262540115050056735206, 27.17809141484929892114680498560, 28.52881146468825999469346907028, 30.0733115480033484527436801700, 30.94146294318809681585321290365, 31.89120861406081866214211640974, 33.25653190038167112890619229099, 34.01172251291200690580101245801

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