Properties

Label 1-95-95.87-r1-0-0
Degree $1$
Conductor $95$
Sign $0.960 - 0.277i$
Analytic cond. $10.2091$
Root an. cond. $10.2091$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.960 - 0.277i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ 0.960 - 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.344419438 - 0.3313005587i\)
\(L(\frac12)\) \(\approx\) \(2.344419438 - 0.3313005587i\)
\(L(1)\) \(\approx\) \(1.530894891 - 0.1959208863i\)
\(L(1)\) \(\approx\) \(1.530894891 - 0.1959208863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + iT \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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