Properties

Degree 1
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.927 + 0.373i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s − 11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + i·23-s + (0.5 − 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)32-s + (0.5 − 0.866i)34-s + ⋯
L(s,χ)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s − 11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + i·23-s + (0.5 − 0.866i)26-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)32-s + (0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.927 + 0.373i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.927 + 0.373i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.927 + 0.373i$
motivic weight  =  \(0\)
character  :  $\chi_{315} (32, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 315,\ (0:\ ),\ -0.927 + 0.373i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.06033475863 + 0.3116489427i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.06033475863 + 0.3116489427i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5037290851 + 0.1793943484i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5037290851 + 0.1793943484i\)

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

−24.81691738065062368115689812960, −24.275936029477748031419381672265, −22.73364112668188304589451374778, −22.10707714451699911690118111036, −20.80146521230657006807077792551, −20.466636978219789761137303583389, −19.321079484019608874849568571825, −18.50284668490545082091836364485, −17.755145577554383265219048080127, −16.81000488195463795851920423067, −15.90790489729107130861798988259, −15.006300979025657748585620351429, −13.559639523652657174025914934910, −12.637798142052239600539916088161, −11.77436614905722972238423611366, −10.66357034637802194439654355325, −9.98914702087436462156537336315, −8.93462512086578721889785564499, −7.8751898528847714211297435305, −7.16117981799885950592499372294, −5.71868416335784632752684426396, −4.33740242732157043748293840454, −2.93808714559449582253573178251, −2.05886355607982253170862122243, −0.25358185061230807462006199457, 1.659650346242510939403172232, 2.85505485518029054316377912632, 4.71883406027589769644770074814, 5.63462391328415859303369079908, 6.936042610067118413013223931586, 7.59589540279136305637225713016, 8.78570696617478129410844593615, 9.57021770289690763585573596383, 10.6430437238123330787689389895, 11.414863167209161025871926015105, 12.748257794915709745461240329333, 13.88575117236056695131703124972, 14.92580025907964047609391069240, 15.73523067725194725599575044913, 16.52753921558350311810781734969, 17.66352247979827638579206653540, 18.102822691234226861206656879227, 19.38617433123705437703407217552, 19.82463093068715706718993023996, 21.03079560676504221244945360084, 21.973841050778924904645628769095, 23.234454765280117474076666567371, 24.07348681771668516842678895690, 24.61219919661122258060853231654, 25.938072604772538421517416372482

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