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Properties

Label 1-91-91.48-r1-0-0
Degree $1$
Conductor $91$
Sign $-0.711 - 0.702i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s − 12-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s − 12-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ -0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2050940551 - 0.4996656788i\)
\(L(\frac12)\) \(\approx\) \(0.2050940551 - 0.4996656788i\)
\(L(1)\) \(\approx\) \(0.6351074874 - 0.1745486873i\)
\(L(1)\) \(\approx\) \(0.6351074874 - 0.1745486873i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
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

−30.923573254233670365427807133122, −29.44324979121013977019338662045, −28.2189449479764107579862352985, −27.28520215526350225106868512051, −26.06356683576215650223864371968, −25.465802347702119438895126214148, −24.12690946909375624690652152805, −23.6077628921590562442125734591, −22.62970411534342363487328803005, −20.48145647632439969729317262515, −19.57350742406606131843201421219, −18.6734647548246072443286176534, −17.79447674331118771986307458603, −16.46290810538119408109006738056, −15.23251280444317794065755214749, −14.50156436847091193906149335144, −13.09811138997169159113567350779, −11.946435219617913186797845287670, −10.24841209457702708946694390714, −8.780682881010945420974591051691, −7.75060931159728590413115958499, −7.121946678414538556676219355457, −5.54884820719730135150978231629, −3.73851423049549854701766582596, −1.559797936415798293389634861768, 0.28260584564158986588211094826, 2.72452133476376661598929120519, 3.71244866499038340168992079995, 4.964374395328413737099253672325, 7.59382256462007068138722594002, 8.50978659001541655773405521837, 9.61508307234554871027330865379, 10.88598224730851055493320620102, 11.62371004841410996737189913169, 13.15487354511426607523447458160, 14.42369050215031008334127985870, 15.926386389796745952815047329457, 16.53075413477629691194749381772, 18.25962046623538113176709842889, 19.287152224841314840543083940170, 20.14013726870933120292900873280, 21.01320372326687910170974199535, 22.07311875991833254721629121566, 23.05509519575281694541544477966, 24.64932636560842202546404322955, 26.24986034265509903875166830233, 26.65370353171126419586440247894, 27.680809327164464268674829444559, 28.40175559689678680464343590858, 29.78124650129052384624522523638

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