Properties

Degree 1
Conductor $ 3 \cdot 29 $
Sign $0.00819 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.00819 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.00819 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $0.00819 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{87} (71, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 87,\ (1:\ ),\ 0.00819 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5884051723 - 0.5932455688i$
$L(\frac12,\chi)$  $\approx$  $0.5884051723 - 0.5932455688i$
$L(\chi,1)$  $\approx$  0.9496648704 + 0.07540864531i
$L(1,\chi)$  $\approx$  0.9496648704 + 0.07540864531i

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

−30.85024098517124743316384607318, −29.63560627863646152078599759978, −28.7060780911719942739223504668, −27.698202626229247234692285546719, −26.63767526362683146136114643184, −25.26613626539109291649083410003, −23.93850607794220373334217165696, −22.98272425293331146362696627486, −22.114576214326998692354314387646, −21.165881381354207934509804803161, −19.89701950490707829627785578532, −18.79424724909622854060117983737, −18.27788201931554710740999034914, −16.1200497240408635823992223079, −14.98466386812304628360386675969, −14.26987855983460403606097046160, −12.525977756331641198519886816044, −11.97647533363562523202278858662, −10.597940149551576778749804011097, −9.60935925398030838155084738376, −7.82093823088393526100310949281, −6.21063546289647760161402657865, −4.88416489588722923179579792199, −3.317388569393063147831897015733, −2.228323118841654095255889641807, 0.302008017784760113629100837869, 3.20882554293817581576107533148, 4.50864109029888962495634049127, 5.5978979455021169735634604102, 7.392929168084611935796461789240, 7.98730900599210774708145448540, 9.62678955681992009080700193501, 11.385768809060424895830484652702, 12.70519613520848636041063077537, 13.45552312809108621277888573834, 14.804309897569043450698918526981, 16.00651337514499385449887351024, 16.68722219856535046248598035915, 17.81579712339528344374509198845, 19.542369684165137370614493148309, 20.56102038035737449212061427592, 21.6778777951714152909193937193, 23.02971699091162662529869972935, 23.74999872057572058864591657822, 24.50987538210153799952687264995, 25.86692813523053010300287028330, 26.7509584296775701993291824075, 27.76291023809591412926990613855, 29.31854060976622982790121368055, 30.24533117186404939578252653431

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