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Properties

Label 1-91-91.88-r0-0-0
Degree $1$
Conductor $91$
Sign $0.564 + 0.825i$
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  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + 9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s − 19-s + (0.5 + 0.866i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + 9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s − 19-s + (0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.416746436 + 0.7475049420i\)
\(L(\frac12)\) \(\approx\) \(1.416746436 + 0.7475049420i\)
\(L(1)\) \(\approx\) \(1.480275767 + 0.5913559350i\)
\(L(1)\) \(\approx\) \(1.480275767 + 0.5913559350i\)

Euler product

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

−30.111567003434645332055045872493, −29.6760581412256614446742891258, −28.36207580792222761335636843424, −26.99955105701943542331515685658, −26.19774230085903690626672159077, −25.04643447669102090651299086547, −23.785870344676766797520027615239, −22.6233221815822852264963149120, −21.47424480773871067924746040152, −20.849699751403509936044224136890, −19.63185808584389387147749217864, −18.70979333040350496299242647859, −17.880964244777962659021410439622, −15.60123879594537243136559136706, −14.72610173722950657120586597020, −13.66480088781946990935857199040, −13.007301080427015204396665057546, −11.30092963531438220724969726030, −10.18283591305966238016318447076, −9.29918526046122057684128127314, −7.66894085391844235419971409167, −6.04931373259129114264067046151, −4.366494305618967868819839591695, −2.92937457628219937692608693705, −2.11791225206003671105047077571, 2.30342286488688634131709542324, 3.995950546828255445929154492411, 5.16533463723868436390471914110, 6.661244601694606540860893023915, 8.19676391262082818970669309976, 8.74304425017871328967106674875, 10.20879454770999193258961332186, 12.580892626736500488431426622894, 13.13123589870587120196439665342, 14.2121155988687000464206222705, 15.31349735134922349780519054073, 16.261976141439401123708515018, 17.455958815341314448815498098, 18.65947358549327828111895011823, 20.20837368248074191005930142830, 21.11231320480945187073893833150, 21.92386315912675631688682124813, 23.675937155318457915773871154711, 24.26210918700640844318644260812, 25.387415818410777981181485884092, 25.98499274823553872782633064153, 27.06440176744473298464818739646, 28.42220761368312932703120716788, 29.86020174475135715459882430440, 30.936542495419035073436867018409

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