Properties

Label 1-91-91.11-r1-0-0
Degree $1$
Conductor $91$
Sign $0.0247 - 0.999i$
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  + (0.866 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0247 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0247 - 0.999i$
Analytic conductor: \(9.77930\)
Root analytic conductor: \(9.77930\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (1:\ ),\ 0.0247 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.354588453 - 2.413625412i\)
\(L(\frac12)\) \(\approx\) \(2.354588453 - 2.413625412i\)
\(L(1)\) \(\approx\) \(1.862686961 - 1.034420398i\)
\(L(1)\) \(\approx\) \(1.862686961 - 1.034420398i\)

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

−30.44918107270036906251933207573, −30.23619598330428421884667691996, −28.26260076713776170473293393850, −26.792716261726600245665355210373, −26.11278166016366161127804714280, −25.158181305054092543503076028913, −24.059679787327324527203900187451, −23.13271972138778461181531763941, −22.05617732338427967512666995223, −20.850145340292348747543049755470, −19.93936165698240094739038546367, −18.856537205101786077271090125235, −17.30521742967340371927118453973, −15.72641248249628193974644630414, −15.12117709501246949201696832283, −14.2708253192773681322128668727, −12.99103205738826521303766915685, −12.00934712030523164784684160198, −10.462686551280651833836739925368, −8.66586030964709771936236611113, −7.58177846886589515596148895557, −6.72312000395772204824777804408, −4.66921775629153369411451296168, −3.63791204183077942482027244695, −2.39365282552624993829823317131, 1.18909397000364976110077798395, 3.04018322803291303399700866372, 3.9176152673565728208141639334, 5.33147040768434265665517854262, 7.17965050208756046967421281975, 8.45228902129731481423695578309, 9.77787080099631030199139090917, 11.26579244880987740096430761699, 12.36213408580379147814540283519, 13.464227572835299082779144698064, 14.40306653472424621924024738003, 15.53153698513825957569117847579, 16.377372016005571435367999452603, 18.78765466207451155370164724059, 19.31657099595061264827821802071, 20.50510597872536611282185505873, 21.05756652739074026696077019249, 22.418385667988586539713058114059, 23.656897528333850594639847968836, 24.432720068152342890604850720640, 25.39054931724027615069637990852, 27.00109416176016234176316307001, 27.677563804736958914574079147278, 29.20977726241823499737320244388, 30.038941688201035931318944348321

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