Properties

Degree 1
Conductor 79
Sign $0.737 + 0.675i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.987 − 0.160i)2-s + (0.0402 + 0.999i)3-s + (0.948 − 0.316i)4-s + (0.799 + 0.600i)5-s + (0.200 + 0.979i)6-s + (0.845 − 0.534i)7-s + (0.885 − 0.464i)8-s + (−0.996 + 0.0804i)9-s + (0.885 + 0.464i)10-s + (−0.919 − 0.391i)11-s + (0.354 + 0.935i)12-s + (−0.200 + 0.979i)13-s + (0.748 − 0.663i)14-s + (−0.568 + 0.822i)15-s + (0.799 − 0.600i)16-s + (0.748 + 0.663i)17-s + ⋯
L(s,χ)  = 1  + (0.987 − 0.160i)2-s + (0.0402 + 0.999i)3-s + (0.948 − 0.316i)4-s + (0.799 + 0.600i)5-s + (0.200 + 0.979i)6-s + (0.845 − 0.534i)7-s + (0.885 − 0.464i)8-s + (−0.996 + 0.0804i)9-s + (0.885 + 0.464i)10-s + (−0.919 − 0.391i)11-s + (0.354 + 0.935i)12-s + (−0.200 + 0.979i)13-s + (0.748 − 0.663i)14-s + (−0.568 + 0.822i)15-s + (0.799 − 0.600i)16-s + (0.748 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.737 + 0.675i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.737 + 0.675i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(79\)
\( \varepsilon \)  =  $0.737 + 0.675i$
motivic weight  =  \(0\)
character  :  $\chi_{79} (37, \cdot )$
Sato-Tate  :  $\mu(78)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 79,\ (1:\ ),\ 0.737 + 0.675i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.271175955 + 1.272733023i$
$L(\frac12,\chi)$  $\approx$  $3.271175955 + 1.272733023i$
$L(\chi,1)$  $\approx$  2.165665211 + 0.5254728029i
$L(1,\chi)$  $\approx$  2.165665211 + 0.5254728029i

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

−30.90726501511577676197459586763, −29.721018671460571388465148848108, −29.07204312239556432111942769939, −27.89037562946934344937360844546, −25.81160438009775345350633232181, −25.00850309681678859829850856335, −24.37885875530664374796067822841, −23.35413042466264226867922509107, −22.208338312879027197653777034882, −20.80386330380969326943662012956, −20.35103728686346399711759638292, −18.46316695094586180986249990488, −17.58071033280072313347970151278, −16.28664846679082750534059343668, −14.73544335351845180458935754377, −13.87772421055880200867935197555, −12.67443638168722964060506715319, −12.14261960795603022928645451299, −10.485036183789396101757998932722, −8.40788850381361582237021796685, −7.40884517827447473099681342838, −5.72220850281347029513706414800, −5.145581884135914260655180357909, −2.76802447347112074936868682342, −1.58212583436840817529763680568, 2.12376665663376228149885289343, 3.582374976813747849785902959441, 4.91516240709800840593087450393, 5.94464047081070299011035917559, 7.62563286414791494827440289484, 9.64801737372460864263356020750, 10.73499936616692172587708709841, 11.47058587365476516610448789889, 13.43258597758990310922569286909, 14.24121594731931501946929685356, 15.11308057000754685443470646619, 16.387377857436868474763059117369, 17.53196806007092143102997151278, 19.242585505827390703074385868240, 20.706502654476353326419228352674, 21.33032037289645002506861299723, 22.02649919401907499882569729484, 23.32004250026116839819413227445, 24.260764192513134545511793024747, 25.857226145370741898328570975436, 26.42063807406388237322468384041, 27.985137940483384026600767331675, 29.02312421551200443201974943086, 30.06342564517603554178173626059, 31.088423838894074877589070521709

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