Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.637 - 0.770i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s i·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)18-s + (0.866 − 0.5i)19-s i·20-s + ⋯
L(s,χ)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + i·6-s + i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)12-s i·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)18-s + (0.866 − 0.5i)19-s i·20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.637 - 0.770i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.637 - 0.770i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.637 - 0.770i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (5, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (0:\ ),\ 0.637 - 0.770i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7657747012 - 0.3604868643i$
$L(\frac12,\chi)$  $\approx$  $0.7657747012 - 0.3604868643i$
$L(\chi,1)$  $\approx$  0.8651577362 - 0.2028903159i
$L(1,\chi)$  $\approx$  0.8651577362 - 0.2028903159i

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

−30.6010293485636659872179331398, −29.13437188810186350300068340366, −28.60075970974641706470454234834, −27.17770385919618291992935625274, −26.55067336173291441490356284926, −25.59956023400253511771855585649, −24.88284401914443322082753847775, −22.70127581955659015025923538266, −21.75009763893202921179089076159, −20.815315300102989333667862443025, −20.1247144208488369304153021725, −18.66049291317440223552968636294, −17.84293614327427426734017085010, −16.567785289957127576545526421677, −15.57132931908963065921872661791, −14.226089647962232353982774223228, −12.98147398595974700884039528719, −11.27368138345296330204364528196, −10.22594536471817715540908411428, −9.58346343908930289341974175994, −8.343386395672777055876003531631, −6.951668043645201690633250233820, −5.03913284562299277258908175247, −3.20074050304890446659437511649, −2.19620943368251717288900547907, 1.25858041899187614885752249552, 2.619316045247704945489546699328, 5.38300089059643205137669271200, 6.45687753349337531030482413280, 7.78069097611022173563075403942, 8.75927948560909455024607451333, 9.77911252264840168802779305697, 11.26084166249690618928916871863, 12.95316143753736937065313625806, 13.81480089038691559265579045635, 15.081725613523206108249118541906, 16.4007591610593073109038250323, 17.623759135015453100908317669511, 18.21357092193568028113428257116, 19.430905196451587086132749219689, 20.32715428772526865032699144219, 21.50872649499381081583439759013, 23.489571176995895220581312669582, 24.225532837203689908588874785837, 25.11314895424931410819903527331, 25.93097453451027384964477963550, 26.8093865774428349645567936112, 28.46504323736611175441827549260, 28.992536596024305642409741567194, 29.96934325606346271953899464608

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