Properties

Degree 1
Conductor $ 3 \cdot 29 $
Sign $-0.314 + 0.949i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.623 − 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $-0.314 + 0.949i$
motivic weight  =  \(0\)
character  :  $\chi_{87} (74, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 87,\ (1:\ ),\ -0.314 + 0.949i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.02520924793 - 0.03490948854i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.02520924793 - 0.03490948854i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4664195493 - 0.2489149754i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4664195493 - 0.2489149754i\)

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

−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