Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.575 - 0.818i$
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 + 3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s i·8-s + 9-s − 10-s i·11-s + (0.5 + 0.866i)12-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)18-s + i·19-s + (0.866 + 0.5i)20-s + ⋯
L(s,χ)  = 1  + (−0.866 − 0.5i)2-s + 3-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s i·8-s + 9-s − 10-s i·11-s + (0.5 + 0.866i)12-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)18-s + i·19-s + (0.866 + 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.575 - 0.818i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.575 - 0.818i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.575 - 0.818i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (72, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (1:\ ),\ 0.575 - 0.818i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.683195415 - 0.8741859850i$
$L(\frac12,\chi)$  $\approx$  $1.683195415 - 0.8741859850i$
$L(\chi,1)$  $\approx$  1.179230880 - 0.3786673472i
$L(1,\chi)$  $\approx$  1.179230880 - 0.3786673472i

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

−30.14103478498659295517288342043, −29.29178852321251989270132357807, −27.983452211592013733390176910986, −26.93588581278610824105296570053, −25.86144687208955807510640545121, −25.45778296320110604887488416028, −24.49334382470293554991991286165, −23.12710328292543678548318885399, −21.56571812962238478592312545573, −20.455939029246232249561339498294, −19.53495804679433863773521199240, −18.3309359592572598332168816812, −17.67510298975612263752631587893, −16.194631359253257779994051503662, −14.977946696563431738298282498400, −14.29749566258765471929580724550, −13.025702114381841383853248085551, −11.02370446434656136000234592654, −9.68202792089305957316051814750, −9.23501896961263679199238723311, −7.59427263297746908817748834030, −6.812088951934033594223579268478, −5.11092531214828776890676783457, −2.83617855173816647576584208689, −1.60070619531675919186821341549, 1.208401090543298173710978276, 2.49358302240487018974813730544, 3.89501226271420509879228659412, 6.095483107376680364286719156581, 7.821366293016444784449007053018, 8.72461489444381165494991039076, 9.67169083687334575074767519505, 10.717860982204149349480112170508, 12.46599174491416353850199869759, 13.35877516586219233142876032412, 14.62428309009654588699374448787, 16.21156908581182029322978992278, 17.064428823462968837392219758919, 18.49064106793515032305264390747, 19.18698567695848857277034511109, 20.47959162822358827085921398870, 21.07884072788471853557590126588, 22.00390392369714609347695956944, 24.209402249131807723924799497233, 25.02819354577760515990985424064, 25.895543670052449707435788153518, 26.78777770035603942338950936764, 27.82642906940397046015504119808, 29.0646559468704359864916465943, 29.79708805031129864228764403519

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