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Properties

Label 1-91-91.6-r0-0-0
Degree $1$
Conductor $91$
Sign $-0.466 + 0.884i$
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 + i·5-s + (−0.866 − 0.5i)6-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + 12-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + (−0.866 − 0.5i)19-s + (0.866 + 0.5i)20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + i·5-s + (−0.866 − 0.5i)6-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + 12-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + (−0.866 − 0.5i)19-s + (0.866 + 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.466 + 0.884i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ -0.466 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3949808709 + 0.6546070416i\)
\(L(\frac12)\) \(\approx\) \(0.3949808709 + 0.6546070416i\)
\(L(1)\) \(\approx\) \(0.6486376873 + 0.4995089310i\)
\(L(1)\) \(\approx\) \(0.6486376873 + 0.4995089310i\)

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 \( 1 + iT \)
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 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + iT \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−29.85205951389989110041378355673, −29.08308730949067253299092191991, −28.048044499136390239487699631882, −27.09207163862886864459768609176, −25.73294051164288088441105036247, −24.97630804836124416415139495942, −24.19539458356301327089508099189, −22.63978959322269840949175592818, −20.99982253484315680181503206913, −20.24195165398374968262555992870, −19.464555201516274305632685711483, −18.34732882753709717687530842578, −17.3131248137140641967833686588, −16.40006597398149250741298638224, −14.76959954284882849896381692849, −13.16143997696675211799433603568, −12.42746730315709601360322008382, −11.37855945557938562549090282474, −9.52746047966954344694938866382, −8.78663547990276813653838726414, −7.68732834350815199026938257843, −6.44355938617908951214504033941, −4.17934590170012096012253204242, −2.403396089244201872591234485964, −1.1002431481772960027495755253, 2.28862305021429335105737176921, 3.88836706999813834586391942034, 5.786708033432352236950797125709, 7.0321853921658548821983898246, 8.425950498489294790375406685102, 9.4084201538693424075169402672, 10.59344324168887480247389984522, 11.31021342943274215361622331034, 13.77327723294292844881595436109, 14.82228013449450735961332896444, 15.4367805699929085685046795298, 16.74369118215074508292837607914, 17.70086634424369438203905843725, 19.24897725266724244112172791617, 19.59972587078246931648138625105, 21.19886760850674783173868873796, 22.18483796413796462305614762141, 23.451219325908894201573929282541, 24.888280335747940678839769925723, 25.780386180529412704081945764982, 26.59072480894586073411784551751, 27.31293424718960354011725058520, 28.26922554783518969380529255268, 29.64428786206047628119180792736, 30.69352277671444217131868531857

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