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Properties

Label 1-87-87.47-r0-0-0
Degree $1$
Conductor $87$
Sign $0.582 - 0.813i$
Analytic cond. $0.404026$
Root an. cond. $0.404026$
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  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (0.623 + 0.781i)5-s + (−0.222 − 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.974 + 0.222i)10-s + (−0.433 + 0.900i)11-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (−0.974 − 0.222i)19-s + (0.900 − 0.433i)20-s + (0.222 + 0.974i)22-s + (−0.623 + 0.781i)23-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (0.623 + 0.781i)5-s + (−0.222 − 0.974i)7-s + (−0.433 − 0.900i)8-s + (0.974 + 0.222i)10-s + (−0.433 + 0.900i)11-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (−0.974 − 0.222i)19-s + (0.900 − 0.433i)20-s + (0.222 + 0.974i)22-s + (−0.623 + 0.781i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.582 - 0.813i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ 0.582 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337856994 - 0.6876338840i\)
\(L(\frac12)\) \(\approx\) \(1.337856994 - 0.6876338840i\)
\(L(1)\) \(\approx\) \(1.427918592 - 0.5219022230i\)
\(L(1)\) \(\approx\) \(1.427918592 - 0.5219022230i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.781 - 0.623i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.433 + 0.900i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.974 - 0.222i)T \)
23 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (-0.781 + 0.623i)T \)
37 \( 1 + (0.433 + 0.900i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (0.433 - 0.900i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (0.974 - 0.222i)T \)
67 \( 1 + (0.900 - 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.781 - 0.623i)T \)
79 \( 1 + (-0.433 - 0.900i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.781 - 0.623i)T \)
97 \( 1 + (0.974 + 0.222i)T \)
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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

−31.05106448392791082877874016317, −29.8761289917923846094571341681, −28.79251979713037757590619982263, −27.74878278992658615027006353096, −26.089526667738668134727742585805, −25.381438709940837823034557476168, −24.4061580161798404327162550990, −23.58504980099336479561854350031, −22.14423149164620382213124997451, −21.40312212365465370306239176735, −20.51877930301598616081673414315, −18.78085512550301913368362751398, −17.52391825540653800945093597760, −16.3969106784373827730038796888, −15.575188852993115434505644219828, −14.273332851305891568783915305746, −13.04000292649973726481879118255, −12.46167420279587410586814857659, −10.83523357938837967283858034371, −8.89772843082360934349354760685, −8.19366253852069930024753333385, −6.0737766291520888911081434840, −5.66822727881280361996438601717, −4.00298706060198382943207245973, −2.339959707431405708491522456496, 1.819311505409790338539611350766, 3.279939814965767265330987142191, 4.62307885699663295491232340662, 6.20128047722847333349182153186, 7.2125317204936584274230080370, 9.54608815239664236118933697434, 10.45448187359479494506273163833, 11.4112118346438036452149770187, 13.020860138792997466831481338244, 13.77333453379902010273930403428, 14.7787030767791365974697468076, 16.06560663522927567614584423137, 17.708815525878063898412755594841, 18.71597407372066977959048398177, 19.992252914439010602245706510, 20.90379272353280494872617585110, 21.92769499064464676571993912260, 23.08145388840774945764240667606, 23.57100812586396077753734160042, 25.26280761417139041505169604027, 26.118529856231928931573835108298, 27.512328957636208701705525880487, 28.745857572824512483652670338019, 29.62719144621572783946983938243, 30.36679091151122078842156131491

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