Properties

Degree 1
Conductor $ 3 \cdot 29 $
Sign $-0.721 + 0.692i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.974 − 0.222i)8-s + (−0.781 − 0.623i)10-s + (−0.974 + 0.222i)11-s + (0.222 + 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s i·17-s + (0.781 + 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯
L(s,χ)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.974 − 0.222i)8-s + (−0.781 − 0.623i)10-s + (−0.974 + 0.222i)11-s + (0.222 + 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s i·17-s + (0.781 + 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.721 + 0.692i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.721 + 0.692i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $-0.721 + 0.692i$
motivic weight  =  \(0\)
character  :  $\chi_{87} (26, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 87,\ (0:\ ),\ -0.721 + 0.692i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3542728196 + 0.8807266890i$
$L(\frac12,\chi)$  $\approx$  $0.3542728196 + 0.8807266890i$
$L(\chi,1)$  $\approx$  0.7553745800 + 0.7030482866i
$L(1,\chi)$  $\approx$  0.7553745800 + 0.7030482866i

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

−30.5559146149552418690614380661, −29.21143308931650183089577392818, −28.222114545118318319215730850424, −27.2934409928680765237649363120, −26.45163683359203150691191355614, −24.47866581327866012080684575341, −23.60813800656553742635940359292, −22.940808776774456325027437955543, −21.48068570754627831342105370035, −20.457044212323616829372341846323, −19.83479415710897676135723430995, −18.58145858638872266808485952558, −17.40753558669617481653298901576, −15.8045823142082642563770832819, −14.747780353804784180658261718811, −13.36170877342382247441106825292, −12.53003184622763307443503758391, −11.13083534946599178701667202623, −10.50119772771181409105468632491, −8.72469479673940643525755678228, −7.57842496795840031996779403808, −5.42643307801173058124215774741, −4.34122125374951387539342939255, −3.06009331908932082598530103317, −0.982978111281199261678279721400, 2.848667571574123231337718239800, 4.40013847907254126495862282986, 5.565937861276482788966325519870, 7.15414324291779301247650582347, 7.999215966436878882566264230806, 9.284557523354463001135296268, 11.30755405435786989461598581007, 12.19077511250837601108800440245, 13.65729432296885366032889237918, 14.82179835625176955632391453725, 15.59976323226412685677486134995, 16.568641092988247143135442200585, 18.22741400886533025441298701494, 18.678722310701522983815328721061, 20.58863458548226820049009144670, 21.63493043932739930777472592838, 22.79768433191037581555066619095, 23.627980800399616680911214943790, 24.55275137374141341570842157320, 25.69134435347347885529358425003, 26.762282525600077721482367613, 27.50411794465355672853204970652, 28.884428474762510969817586230222, 30.52871386700763659797827243470, 31.31979488357886275912369330146

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