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Properties

Label 1-91-91.5-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: $\chi_{91} (5, \cdot )$
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

−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