Properties

Degree 1
Conductor $ 3 \cdot 7 \cdot 191 $
Sign $-0.199 + 0.979i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.984 − 0.175i)2-s + (0.938 + 0.345i)4-s + (0.716 + 0.697i)5-s + (−0.863 − 0.504i)8-s + (−0.583 − 0.812i)10-s + (0.998 + 0.0550i)11-s + (−0.115 + 0.993i)13-s + (0.761 + 0.648i)16-s + (0.224 + 0.974i)17-s + (−0.583 + 0.812i)19-s + (0.431 + 0.901i)20-s + (−0.973 − 0.229i)22-s + (0.949 − 0.314i)23-s + (0.0275 + 0.999i)25-s + (0.287 − 0.957i)26-s + ⋯
L(s,χ)  = 1  + (−0.984 − 0.175i)2-s + (0.938 + 0.345i)4-s + (0.716 + 0.697i)5-s + (−0.863 − 0.504i)8-s + (−0.583 − 0.812i)10-s + (0.998 + 0.0550i)11-s + (−0.115 + 0.993i)13-s + (0.761 + 0.648i)16-s + (0.224 + 0.974i)17-s + (−0.583 + 0.812i)19-s + (0.431 + 0.901i)20-s + (−0.973 − 0.229i)22-s + (0.949 − 0.314i)23-s + (0.0275 + 0.999i)25-s + (0.287 − 0.957i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.199 + 0.979i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.199 + 0.979i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4011\)    =    \(3 \cdot 7 \cdot 191\)
\( \varepsilon \)  =  $-0.199 + 0.979i$
motivic weight  =  \(0\)
character  :  $\chi_{4011} (17, \cdot )$
Sato-Tate  :  $\mu(570)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4011,\ (0:\ ),\ -0.199 + 0.979i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8325009969 + 1.018895614i$
$L(\frac12,\chi)$  $\approx$  $0.8325009969 + 1.018895614i$
$L(\chi,1)$  $\approx$  0.8305181313 + 0.2434248557i
$L(1,\chi)$  $\approx$  0.8305181313 + 0.2434248557i

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

−18.1243620918949515556997069363, −17.538170051092040136952836134454, −17.0054178775920161667752150817, −16.52449218394187009558008480332, −15.71890156568880317033783765339, −15.00050694037750630035109307443, −14.36431488477129411079169197083, −13.47862942122852206946136851666, −12.71713695242153679433267288305, −12.09884751256608119476355180235, −11.11686700438295695942195599888, −10.736977883904492860033814610173, −9.66121951778963715355323359122, −9.17811445153294068126101483652, −8.89798740500507495027420751362, −7.75838181290078022384918394325, −7.26239509105717184082024736097, −6.34901668747400367524211663054, −5.66153840000129225106120843722, −5.07039227675692393243227840530, −3.9637901558967411283983303689, −2.83457243044860794142291466735, −2.17800529787116099711988511687, −1.168264294132551736415001578616, −0.56078535270114219458986552838, 1.20903293806893852533714900073, 1.81502254915682128361715190993, 2.50308116295085702253798757989, 3.54203610401267790149148592888, 4.12548284638577795314091118868, 5.53949668741598102496952685302, 6.280742370813911580334303160626, 6.757261752683084792469871565365, 7.43538842714480114857219689596, 8.40005362952274196739568185951, 9.06884695451677719160080987548, 9.683066724430371296090933217047, 10.23222629860679526095546894668, 11.22361895788562521426293639154, 11.32962316190906511625214064121, 12.51589710878498695230621613274, 12.96709994890483995664775460137, 14.13505407434852079253675627164, 14.77397036803712059414108286524, 15.03739657590029261940613074182, 16.47036629442165859433412755762, 16.654516785797125070599702459451, 17.34576457790999478350765271046, 17.98721590196052610236873254492, 18.797833500187054702860645495575

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