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  :  $\chi_{95} (49, \cdot )$
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

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

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