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Properties

Label 1-91-91.73-r0-0-0
Degree $1$
Conductor $91$
Sign $0.637 + 0.770i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
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  + (−0.866 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7657747012 + 0.3604868643i\)
\(L(\frac12)\) \(\approx\) \(0.7657747012 + 0.3604868643i\)
\(L(1)\) \(\approx\) \(0.8651577362 + 0.2028903159i\)
\(L(1)\) \(\approx\) \(0.8651577362 + 0.2028903159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 - iT \)
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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

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

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