Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $-0.671 - 0.740i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s − 7-s − 8-s + (−0.5 − 0.866i)9-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + (−0.5 + 0.866i)21-s + (0.5 − 0.866i)22-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s − 7-s − 8-s + (−0.5 − 0.866i)9-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 18-s + (−0.5 + 0.866i)21-s + (0.5 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.671 - 0.740i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.671 - 0.740i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $-0.671 - 0.740i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (64, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 95,\ (0:\ ),\ -0.671 - 0.740i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5228970179 - 1.179861651i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5228970179 - 1.179861651i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9177439011 - 0.9417770619i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9177439011 - 0.9417770619i\)

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

−30.84491014627610342653236084454, −29.88825001605465705447602872552, −28.15821971373866719577151029183, −27.23947900968980724808772030380, −26.115602647358281627958642063940, −25.50826530058006954680523496480, −24.55206526466649149219926218420, −22.92620465248974627870070736218, −22.400376307061993162315468901724, −21.333367182404571580267970770825, −20.168312657579990913112481408143, −18.95466079263623328417499419820, −17.240000964013814086249883244570, −16.40785034518957818681618069046, −15.43442627450889253563110671995, −14.564623696431030316508439949278, −13.4389128734075009573412254801, −12.31284408276088294332005466707, −10.491856211829518165136093476080, −9.21384484575818812090879747829, −8.28009902937974379752075157865, −6.69768439592008731741439859645, −5.49618933285189871085703651262, −3.98263028195598902515093218485, −3.1328440918878846592268258955, 1.296923330494725049029294919025, 2.83952056973052612580228467360, 3.944669071698011407123983561689, 5.94135634519053757600862516734, 6.989654342882925759241790926646, 8.92231891657506949402537761569, 9.69348740708964302684551174605, 11.50461000417158423687801756630, 12.27612072139091115288707950588, 13.49982687410953130145814621061, 14.08137989536986348306958312075, 15.4767701427701453221383025711, 17.16341706343034806031535889036, 18.68309814610733979929367930204, 19.197088708469135759285614073191, 20.15019445008209247633985769022, 21.23249625982532991277639728290, 22.58733812510470909383998418320, 23.30171340078827647152503981196, 24.4964474333522027818424570221, 25.49334583912964409728507953249, 26.72042037582687829703450264991, 28.10455416886462306958010631294, 29.141104109322435021123000749499, 29.8133426825595782339532768477

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