Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $0.624 + 0.780i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 18-s + (0.939 − 0.342i)21-s + (0.173 − 0.984i)22-s + ⋯
L(s,χ)  = 1  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 + 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 18-s + (0.939 − 0.342i)21-s + (0.173 − 0.984i)22-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.624 + 0.780i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.624 + 0.780i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $0.624 + 0.780i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (29, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (1:\ ),\ 0.624 + 0.780i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9290411688 + 0.4463925469i$
$L(\frac12,\chi)$  $\approx$  $0.9290411688 + 0.4463925469i$
$L(\chi,1)$  $\approx$  0.7771671808 + 0.07758658242i
$L(1,\chi)$  $\approx$  0.7771671808 + 0.07758658242i

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

−29.71880757382283544365850611438, −28.46942503005971554000651886550, −27.555885661174959442862675487723, −26.80235153071861562680056517873, −26.05952177020344839207985079112, −24.92131738571691288222474252710, −23.50099123303605390131785330194, −22.055797183732882101645115012850, −20.979949829298519251432116789405, −20.388437003004347270405358938504, −19.28884272461440731228198556858, −17.937234665844706126613919781770, −16.86825426808500863446325516036, −16.0850559689298936402092553226, −14.87352271757780502359869290567, −13.474686770567993444519672079781, −11.75638334919090956413217047778, −10.55771622041542564414227818994, −10.12212972520178229595769097618, −8.50227447118137286522556636307, −7.795482352239347116468203307510, −5.82557344806491324866721551023, −4.04350697501726152293043564885, −2.808465329216085950946789462173, −0.65866254621464409718261893224, 1.42550695400141056749037600792, 2.54467343083316768704561601172, 5.3067891655173060757304439237, 6.58502890864581983620781618586, 7.7048481772197433553379445002, 8.633407191238348213662731911923, 9.84694199342622915256678593237, 11.5242735982664726238167862903, 12.27010191446990908290445773917, 14.01001825406222704214576979769, 14.98597419640962542176289683363, 16.248033565916484488703733562789, 17.62560814567443355266868707280, 18.2876176287608750384917585599, 19.12081535647369985991326849522, 20.23379661507116250908859486634, 21.37308051862811368514996714264, 23.30312576322559787046777067785, 23.95930950640067444227095785474, 25.2076205649955216449185658736, 25.57962055758632010284312028012, 26.9036832230336598162581584719, 28.23946018200689080215448421191, 28.731409640822660768760028931719, 29.989769428922358878798569971523

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