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Properties

Label 1-87-87.74-r1-0-0
Degree $1$
Conductor $87$
Sign $-0.314 + 0.949i$
Analytic cond. $9.34944$
Root an. cond. $9.34944$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.623 − 0.781i)5-s + (−0.222 − 0.974i)7-s + (0.900 − 0.433i)8-s + (−0.222 + 0.974i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.900 − 0.433i)16-s − 17-s + (−0.222 + 0.974i)19-s + (0.900 − 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.623 + 0.781i)23-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.623 − 0.781i)5-s + (−0.222 − 0.974i)7-s + (0.900 − 0.433i)8-s + (−0.222 + 0.974i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (−0.623 + 0.781i)14-s + (−0.900 − 0.433i)16-s − 17-s + (−0.222 + 0.974i)19-s + (0.900 − 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.623 + 0.781i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.314 + 0.949i$
Analytic conductor: \(9.34944\)
Root analytic conductor: \(9.34944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (1:\ ),\ -0.314 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02520924793 - 0.03490948854i\)
\(L(\frac12)\) \(\approx\) \(0.02520924793 - 0.03490948854i\)
\(L(1)\) \(\approx\) \(0.4664195493 - 0.2489149754i\)
\(L(1)\) \(\approx\) \(0.4664195493 - 0.2489149754i\)

Euler product

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

−31.35551827899524706417047874118, −30.02408177328747002104696814927, −28.70692051175002895733326978449, −27.75829681093649040328427220642, −26.75543920455286549512908297952, −26.02369898969218970196923602622, −24.71164568724329430597361714256, −24.05362733297392538123068549223, −22.54640376629760123934196105350, −21.96421929083576514775311761077, −19.79579490369800240854236657934, −19.14630975329466923548013611179, −18.17223573112285801862716128250, −16.99563518091275256205653401520, −15.70582857647960298755385480406, −14.99506764452870115524516492418, −13.92933460584976931442839531344, −12.0209420975658486997144830733, −10.915035010611035925619107434217, −9.44715742809788339971293941795, −8.47202340151848800072818254432, −7.02623267553460125407488641690, −6.17981264315517635700322131957, −4.4577338408889481548355486309, −2.41335052835512465540559408107, 0.02480869732020586022428232244, 1.55170639637396704477006952253, 3.60606077527540798155459915516, 4.59807198155359088739957847086, 7.02933818466844817868357117878, 8.11819981366497644887299171209, 9.37554606351179874589483129392, 10.44577686008030586965291803734, 11.82282809820243075817389663844, 12.61606141309812691513852864168, 13.89286251735049313852384559263, 15.679977380591294654783634383382, 16.93939721835834196868459337500, 17.48430629597527427844131685478, 19.22526361919582052512365801628, 19.945304607537929260273029004568, 20.58786708265564132397799196376, 22.08056980276029084297116714260, 23.064897010144470850787848409596, 24.427325062206465916929349947828, 25.58455296413980942745978019804, 26.99854388519954550398832140274, 27.37556206641131979981818886175, 28.60933439540903226298424884455, 29.543757555484756610807778110986

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