Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.728 - 0.684i$
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 + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + i·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 + i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s i·20-s + ⋯
L(s,χ)  = 1  + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)5-s + i·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 + i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)18-s + (0.866 − 0.5i)19-s i·20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.728 - 0.684i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.728 - 0.684i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.728 - 0.684i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (86, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (1:\ ),\ 0.728 - 0.684i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.614849474 - 1.035659125i$
$L(\frac12,\chi)$  $\approx$  $2.614849474 - 1.035659125i$
$L(\chi,1)$  $\approx$  1.751839718 - 0.4113387713i
$L(1,\chi)$  $\approx$  1.751839718 - 0.4113387713i

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

−30.2726897329830076627029807510, −29.545546502660199355429771628732, −28.64977100863109301498541901064, −26.89951103121706809067520043499, −25.5955642370374034965558032154, −24.895236359966842340725332778081, −24.01459642463311021001696821660, −22.81280614071290520370257912285, −22.14223685581555231755170824025, −21.092770861302484655033700334944, −19.520084911669578087074244016601, −18.20950249957706883179354627775, −17.20936109909102026230907264954, −16.38398120893321858943283484233, −14.58124489210666405484845888769, −13.93066700211380984149139204721, −12.81774605109367203522031978511, −11.79679845850160082239455964967, −10.55873312363277819677685874429, −8.54162933220537183350946953898, −7.07517932619632986271185809181, −6.2379959482315684775056417625, −5.27999506743889555971022209047, −3.29787466036230658081505613353, −1.71008699301732976739282661570, 1.23452940176342957758022581948, 3.140294319917698542125760862020, 4.64378172183826716240834747817, 5.45115154909626713222807608181, 6.709065333580204594052797679658, 9.31971212702095408478168020625, 9.9124368048430713667409145597, 11.33693806100173321330549310668, 12.23569673448689109806984448560, 13.60579584300881016494404491119, 14.61166403563050044867242974938, 15.817202036163883584410731671818, 16.88774659789588489759338221948, 18.07248587681699622807267933456, 19.87292481757624562276060826228, 20.69226697430050716408920009854, 21.61844751776850909027400445256, 22.37918449425610295998153688160, 23.36947576318136044509262684117, 24.660866722218069523811969904819, 25.61213907427043054629641784599, 27.28890565566637860778400900304, 28.15508108855561127542434086473, 29.0469822174209261086585159984, 29.8303547769367453990593908150

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