Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $0.845 + 0.533i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.845 + 0.533i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.845 + 0.533i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $0.845 + 0.533i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (24, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 95,\ (0:\ ),\ 0.845 + 0.533i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9398706067 + 0.2715728501i$
$L(\frac12,\chi)$  $\approx$  $0.9398706067 + 0.2715728501i$
$L(\chi,1)$  $\approx$  0.9711185436 + 0.2403333339i
$L(1,\chi)$  $\approx$  0.9711185436 + 0.2403333339i

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

−30.28903844525284651030229136206, −28.924934422468191092516807476046, −28.11325515933195740608134640793, −26.95859523148855408773325357765, −25.96349172518357766324328934323, −25.21365622120182109876122672198, −24.19412134872849662170997278561, −22.49946780374511918975377623114, −21.06682031404766613594235927905, −20.6337345064827198194405066263, −19.43153064219098321138673010961, −18.34818824434334560081461465648, −17.9006710707862972932836377997, −16.0662243727242049694642614291, −15.05262767817595542318966495802, −13.55669120763676759857589463150, −12.51566265918008443172662491001, −11.40836842034528263468777100858, −9.92409096911201095284465679147, −8.83376449997962239896239939936, −8.061326839581362901472910714738, −6.72914088820985498738093310352, −4.39604106800745702544454008547, −2.75499306371509766508661165391, −1.781918095424741201044919671263, 1.59318682945951403135706064714, 3.57624278820693916662090787617, 5.19274537513293561938117087424, 6.87967715361692860347076227819, 8.128462746416670406499429447750, 8.760558005253667685418250992376, 10.29752261999715978574497880716, 10.9657039535533141687097000865, 13.42367971616142884971824908002, 14.10862409263437016671922605404, 15.39397251618683506523397737986, 16.152913253466393382579381845379, 17.44183526655684360950897335081, 18.55885374896452736427118631047, 19.674898128549066750350143525282, 20.46165552794910168816674527135, 21.617479066342630354877493899905, 23.42619605420842368233325533072, 24.17124012986115197345622417976, 25.34709634622700110931144559498, 26.20536764841045111996248603327, 26.96702813027569224056367961806, 27.768708591573862931623233062181, 29.14721766861342294081216617781, 30.32430211765681752144216858519

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