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Properties

Label 1-91-91.32-r1-0-0
Degree $1$
Conductor $91$
Sign $-0.810 + 0.585i$
Analytic cond. $9.77930$
Root an. cond. $9.77930$
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  − i·2-s + (−0.5 + 0.866i)3-s − 4-s + (0.866 + 0.5i)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 + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)15-s + 16-s − 17-s + (−0.866 + 0.5i)18-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯
L(s)  = 1  − i·2-s + (−0.5 + 0.866i)3-s − 4-s + (0.866 + 0.5i)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 + (0.5 − 0.866i)12-s + (−0.866 + 0.5i)15-s + 16-s − 17-s + (−0.866 + 0.5i)18-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.810 + 0.585i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ -0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05869203956 + 0.1814782838i\)
\(L(\frac12)\) \(\approx\) \(0.05869203956 + 0.1814782838i\)
\(L(1)\) \(\approx\) \(0.6552341662 - 0.06705931931i\)
\(L(1)\) \(\approx\) \(0.6552341662 - 0.06705931931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + iT \)
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 \( 1 + iT \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + iT \)
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

−29.66689929626299795471060387404, −28.56989324654164602628192441104, −27.83332369365840110289963198663, −26.10550068778038237472174923686, −25.46788011848396167382537691262, −24.32011218899206658899617206333, −23.79422837334263951790178345786, −22.56620869298784662680725506336, −21.58458934238405624784995089900, −19.95487759091627201811067902319, −18.42406566094387493101729601493, −17.778428967390603566427179701438, −16.917482136411043545920082130367, −15.801635171865661254602348346173, −14.3022111714943631248655503216, −13.17237052012638565192073204464, −12.651236569064651225632927768971, −10.69510645988850152503539754187, −9.22191261188131321313625370440, −8.00402540986125018962074280204, −6.77600960462081145046362209745, −5.74634702209193737058700465388, −4.70638813813313373026618446944, −2.01213560899659944580508452267, −0.08712184384855611536598193592, 2.18681694169503596206561855607, 3.60730460450412587632035526026, 5.03959730997008992405959367818, 6.17165255383881319088257178473, 8.51323396461322154630579650079, 9.787314891306898123400394928779, 10.54346636805817758192957172497, 11.42177822544112101657810296874, 12.91397073834300565326596748443, 14.020118843125880498019830394739, 15.24443013529347885648972960838, 16.78854153185745716936344464356, 17.810968694118304425513075943203, 18.67532682793129242572873138396, 20.219970823577848967164268175882, 21.20718199061186132797905261265, 21.86340195896712282081490980938, 22.71883557518867341918835442235, 23.8862776695565057166722207507, 25.82998607314284064833006008858, 26.58307110271856631600923534644, 27.56155922974310473354160123328, 28.76830144049808549723599947962, 29.21047303594523537215034942023, 30.35093798493584504007095100887

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