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Properties

Label 1-91-91.33-r0-0-0
Degree $1$
Conductor $91$
Sign $-0.100 - 0.994i$
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 − 3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s − i·8-s + 9-s + 10-s − i·11-s + (−0.5 − 0.866i)12-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)18-s − i·19-s + (−0.866 − 0.5i)20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s − i·8-s + 9-s + 10-s − i·11-s + (−0.5 − 0.866i)12-s + (0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)18-s − i·19-s + (−0.866 − 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ -0.100 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2266624442 - 0.2507845390i\)
\(L(\frac12)\) \(\approx\) \(0.2266624442 - 0.2507845390i\)
\(L(1)\) \(\approx\) \(0.4218083866 - 0.1453174363i\)
\(L(1)\) \(\approx\) \(0.4218083866 - 0.1453174363i\)

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 \( 1 - T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 - iT \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 - T \)
67 \( 1 + iT \)
71 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
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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

−30.53574558943461611391007759688, −29.19549264412844883102296019209, −28.32612578532875601350475498785, −27.6454303889867777094480212182, −26.81354817346917782502365951476, −25.436654935142074783209571781701, −24.26724988576062458723429308266, −23.5035414871175391270176648982, −22.64773683119620452043398477877, −20.92284375396121975431570819497, −19.76503543844606018668732453252, −18.72526443240638184937308373414, −17.5749382772466743832783809090, −16.77551593156854733028793519644, −15.76025790855635159249012028453, −14.92472259554428430781780266526, −12.79670775685329336209916180804, −11.712760805632240393812291811967, −10.655339355305079885441167492363, −9.4428134797551405865883066328, −7.96883935175963794405629157002, −6.97166915078106274813127587019, −5.61047068534321134305033041001, −4.321359521966403997775401361251, −1.45925254485628420937393099144, 0.578629855156341805400157712072, 2.86984921752281693248961126814, 4.4365758447132686764854341688, 6.43053284555456787938630270453, 7.43682175531797197262170642761, 8.825591436746465926940905962078, 10.371984405160754425048854442563, 11.275047588119404572117759829212, 11.88920225293737048728704880434, 13.32054516818463415463708836739, 15.420000295157763319431309643534, 16.244239288986045683905766015586, 17.29807694329131897996252674535, 18.47643842854542243760688471791, 19.09015089855765445161530848157, 20.39379171665781765344994459187, 21.741633495532848223810305972857, 22.51478228525711011062262103607, 23.78950898923428797312363415376, 24.85645836427665988338630323981, 26.64713375963792878590727913269, 26.92394062999619330729668873761, 28.09948303933160854698951188237, 28.92346715321668857611293023134, 29.96169655554313894486809326661

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