Properties

Degree 1
Conductor $ 7 \cdot 13 $
Sign $0.372 + 0.927i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)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 + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s − 17-s + (−0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯
L(s,χ)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)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 + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)15-s + 16-s − 17-s + (−0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.372 + 0.927i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.372 + 0.927i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $0.372 + 0.927i$
motivic weight  =  \(0\)
character  :  $\chi_{91} (68, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 91,\ (1:\ ),\ 0.372 + 0.927i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.185169817 + 2.152677575i$
$L(\frac12,\chi)$  $\approx$  $3.185169817 + 2.152677575i$
$L(\chi,1)$  $\approx$  2.201849028 + 0.8897386131i
$L(1,\chi)$  $\approx$  2.201849028 + 0.8897386131i

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

−30.14323409487851962605475455765, −29.00836885419353172956241744854, −28.512329725298044463014254907321, −26.435751059244444170344574875881, −25.14348193379650195796396479640, −24.80107067538935635709829793928, −23.67071212076722136881394906350, −22.79389418205148003838985170998, −21.25131893580809716462670937615, −20.46660400366812620087533464274, −19.65083428305208075228529234089, −18.10359443200615760068803225816, −16.93286383364867861598435199314, −15.54507744765648408278361935759, −14.408110051760002344323523894415, −13.24098635216284888275285419524, −12.761119515408858404935414234993, −11.58554123658822047014260413099, −9.784429838982925856811192568208, −8.25471272790176430742713164309, −7.02488342849972649888469566433, −5.76150655215480074085753722798, −4.43968367265399756029011062582, −2.65973404623887319990438868594, −1.45192039226743102128059683980, 2.4517720946516190585517525120, 3.33940627049138566889065664762, 4.82289267221164018911124261548, 6.044245047082554872105438471456, 7.47131727576953797727931400384, 9.17796425479932213727581884040, 10.683656621625631311265756363557, 11.21024330242407472520948579449, 13.24596072516263036371477128258, 13.94594931374376557755212672838, 15.04980143752467422099538047690, 15.796881003737518329710841990615, 17.09901653871938785204110978337, 18.811102248824328253918665568039, 20.01433102596633927265335909602, 21.08862375806505150856754290570, 21.924189974655709448110963742645, 22.53444257613548027950542883059, 23.97185255193866820158581324015, 25.13816280095355668222057229669, 26.12519653283424559777193100263, 26.864447276891699556983964372963, 28.529415066925859400314164943802, 29.4729312121865532616338879916, 30.64398159855833291761241882882

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