Properties

Degree 1
Conductor 107
Sign $0.332 + 0.943i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.430 + 0.902i)2-s + (−0.956 + 0.292i)3-s + (−0.630 − 0.776i)4-s + (0.889 + 0.456i)5-s + (0.147 − 0.989i)6-s + (0.263 − 0.964i)7-s + (0.972 − 0.234i)8-s + (0.829 − 0.558i)9-s + (−0.794 + 0.606i)10-s + (0.937 + 0.348i)11-s + (0.829 + 0.558i)12-s + (−0.998 + 0.0592i)13-s + (0.757 + 0.652i)14-s + (−0.984 − 0.176i)15-s + (−0.205 + 0.978i)16-s + (−0.320 + 0.947i)17-s + ⋯
L(s,χ)  = 1  + (−0.430 + 0.902i)2-s + (−0.956 + 0.292i)3-s + (−0.630 − 0.776i)4-s + (0.889 + 0.456i)5-s + (0.147 − 0.989i)6-s + (0.263 − 0.964i)7-s + (0.972 − 0.234i)8-s + (0.829 − 0.558i)9-s + (−0.794 + 0.606i)10-s + (0.937 + 0.348i)11-s + (0.829 + 0.558i)12-s + (−0.998 + 0.0592i)13-s + (0.757 + 0.652i)14-s + (−0.984 − 0.176i)15-s + (−0.205 + 0.978i)16-s + (−0.320 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.332 + 0.943i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.332 + 0.943i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(107\)
\( \varepsilon \)  =  $0.332 + 0.943i$
motivic weight  =  \(0\)
character  :  $\chi_{107} (9, \cdot )$
Sato-Tate  :  $\mu(53)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 107,\ (0:\ ),\ 0.332 + 0.943i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5878487496 + 0.4161656256i$
$L(\frac12,\chi)$  $\approx$  $0.5878487496 + 0.4161656256i$
$L(\chi,1)$  $\approx$  0.6798902140 + 0.3335542338i
$L(1,\chi)$  $\approx$  0.6798902140 + 0.3335542338i

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

−29.23461742603054088560607315868, −28.59981614413501094042285236267, −27.671132087700510720055156173741, −26.809868837127558901314326929246, −24.98298490766237596022746336011, −24.64401516484030613085861142076, −22.825222852006284550698844539417, −21.894323614615946550611811819, −21.4240236521615277855604722999, −19.9891991041294337254249965403, −18.80649433648249338681454454503, −17.83667014437612584028423809337, −17.22123004373885130387820525428, −16.1217104237720961311117034688, −14.14064835073313950953136110453, −12.94729597214855433555859585250, −11.9964011113020380332518523281, −11.26954814920514185843388269364, −9.73910232027026015519473556058, −9.04925515038774609844603782591, −7.33591343756439927517888812127, −5.68672382553171341715455645961, −4.71560249562422398606003842566, −2.542189088152522593996921856774, −1.22127061519362479953674042217, 1.36280321863161345966842563105, 4.21394862948259348157115491145, 5.38388673930203729829917920808, 6.59159697451269578673221292778, 7.3112633676988767628715986996, 9.29421577630795950293624576669, 10.14260947276003265487807371256, 11.05886715044781640314920837995, 12.81801229217351456872542899055, 14.204894518821306369699022322845, 14.95310057104193295014633682393, 16.511952551114600108996400736823, 17.23714071365070902494443318869, 17.73227292142372023710843417990, 19.026369979253085300585902708359, 20.46701411451596537739413076623, 22.12904632109889323591259380089, 22.48758172520201883112089039092, 23.82710899811254714669089400020, 24.57517902019308549315595334105, 25.868109851687619267429789441212, 26.82062628064702416185949807463, 27.49004618134333557107388055615, 28.78095132889219527044784787668, 29.48557324700791613763025686509

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