Properties

Degree $1$
Conductor $315$
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

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.927 - 0.373i$
Motivic weight: \(0\)
Character: $\chi_{315} (128, \cdot )$
Sato-Tate group: $\mu(12)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ -0.927 - 0.373i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.06033475863 - 0.3116489427i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.06033475863 - 0.3116489427i\)
\(L(\chi,1)\) \(\approx\) \(0.5037290851 - 0.1793943484i\)
\(L(1,\chi)\) \(\approx\) \(0.5037290851 - 0.1793943484i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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