Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.0247 + 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.0247 + 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (58, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (1:\ ),\ 0.0247 + 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.354588453 + 2.413625412i$
$L(\frac12,\chi)$  $\approx$  $2.354588453 + 2.413625412i$
$L(\chi,1)$  $\approx$  1.862686961 + 1.034420398i
$L(1,\chi)$  $\approx$  1.862686961 + 1.034420398i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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