Properties

Degree 1
Conductor $ 3^{2} \cdot 13 $
Sign $0.815 - 0.578i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + 11-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + 22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯
L(s,χ)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + 11-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + 22-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.815 - 0.578i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.815 - 0.578i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(117\)    =    \(3^{2} \cdot 13\)
\( \varepsilon \)  =  $0.815 - 0.578i$
motivic weight  =  \(0\)
character  :  $\chi_{117} (22, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 117,\ (0:\ ),\ 0.815 - 0.578i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.647660323 - 0.5253185935i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.647660323 - 0.5253185935i\)
\(L(\chi,1)\)  \(\approx\)  \(1.626611814 - 0.3048064331i\)
\(L(1,\chi)\)  \(\approx\)  \(1.626611814 - 0.3048064331i\)

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

−29.55973354326705430174512278275, −28.51018826195320559399662234292, −27.30928950668749952766659173202, −26.00284321324701401005907650395, −25.13497411661965659976247721233, −24.15074763951952062980694544373, −22.9092487114442818774631565547, −22.28058137833218788849357382325, −21.53761144472901597100569564441, −19.9904666093900707562750734914, −19.2919259018650777126081182483, −18.0385942196396164184941951042, −16.40521918882784541990604923330, −15.46230637307917012974217177478, −14.66534880335549437098627710877, −13.601865444899091371439651541519, −12.2041870839215931604851324676, −11.56070342216743678776165403466, −10.30852764955445230256492723007, −8.68277230886967752193625208309, −6.95494350551748548554991332071, −6.355082216406049085171201139457, −4.75914659887944455455381294469, −3.41339982727858896871146016116, −2.365101615483467487660556627835, 1.550931693334795453524122655548, 3.68876591240343172823050680437, 4.28590970271190754365622873290, 5.855970794160857870975365405645, 7.02734097241849911210185459537, 8.31379625477152107681806152691, 9.93786101468673537721310290518, 11.27788147775694386928862756190, 12.33585001552931758315713296040, 13.19179589135443067785005374704, 14.26489918132470015313309601733, 15.46553584790734188436127399810, 16.51180737929382182098113005025, 17.19465014536954908951238315766, 19.44287928572404724277406962631, 19.874037599513999628532804262852, 20.93565247999767526204804538285, 22.06550050186118942205651947912, 23.142944628049649465826873479272, 23.826945944068802240878935427574, 24.81183118270475863204545684974, 25.80326797115326375886656138987, 27.165512812999879065377936898219, 28.275740804171368500687118574161, 29.33883471426740245140215257500

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