Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $0.960 - 0.277i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.960 - 0.277i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.960 - 0.277i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $0.960 - 0.277i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (87, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (1:\ ),\ 0.960 - 0.277i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.344419438 - 0.3313005587i$
$L(\frac12,\chi)$  $\approx$  $2.344419438 - 0.3313005587i$
$L(\chi,1)$  $\approx$  1.530894891 - 0.1959208863i
$L(1,\chi)$  $\approx$  1.530894891 - 0.1959208863i

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

−30.16489505498409524583662232578, −29.3401318075638140198427602883, −27.981285507662123871351668631698, −26.80197816109177471565749516169, −25.436278628090216527021583873395, −24.593617993495818067384258889846, −23.36928303504215627055079046013, −23.0345714216326094941681301290, −21.883875808542308091170760902833, −20.69787557211250435122865080065, −19.42085836040698933478529769584, −17.8318343423190235208002282894, −16.92220743887085390905351057112, −16.20455551278781698767545901776, −14.65963158739122755866810327379, −13.56173982606388225272537685315, −12.64255833897336358220321008447, −11.51038197005386231519810905696, −10.46729554360604336909271778447, −8.24396773145668724696823650862, −7.00484947825168583593965178553, −6.17939700814652310756174467151, −4.82509138828008121652765113998, −3.527293208268228799076109774867, −1.22867737961521653048319483706, 1.28284886304653415904873776766, 3.25373417644507285848385261437, 4.58029910905898443431811046602, 5.74515258895035219731240087968, 6.64765891097233932435973609947, 9.067276273419899540744220282655, 10.17900312670447485889456157719, 11.74383352604032963593422917842, 11.81684461026534567204074578014, 13.430549687604880533960092006620, 14.809850487772893375517730551136, 15.68545338970137126307737387232, 16.786888512355121841634831465058, 18.3197527112816743213111982456, 19.29504059205155820790800581571, 20.90022383506525959187994462266, 21.47211979970008955392067509761, 22.53240176337206722193823406589, 23.21034820959238071699953561556, 24.43187529166938330237808476934, 25.44258953699070675360190227513, 27.26809814722584989004736062668, 28.06857867975249960194055445259, 28.82778208500985338769309299782, 29.86606534623725855144953085660

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