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

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