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Properties

Label 1-79-79.47-r1-0-0
Degree $1$
Conductor $79$
Sign $0.737 - 0.675i$
Analytic cond. $8.48972$
Root an. cond. $8.48972$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.737 - 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.271175955 - 1.272733023i\)
\(L(\frac12)\) \(\approx\) \(3.271175955 - 1.272733023i\)
\(L(1)\) \(\approx\) \(2.165665211 - 0.5254728029i\)
\(L(1)\) \(\approx\) \(2.165665211 - 0.5254728029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.987 + 0.160i)T \)
3 \( 1 + (0.0402 - 0.999i)T \)
5 \( 1 + (0.799 - 0.600i)T \)
7 \( 1 + (0.845 + 0.534i)T \)
11 \( 1 + (-0.919 + 0.391i)T \)
13 \( 1 + (-0.200 - 0.979i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (0.278 + 0.960i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.428 + 0.903i)T \)
31 \( 1 + (-0.632 + 0.774i)T \)
37 \( 1 + (-0.692 + 0.721i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (0.919 + 0.391i)T \)
47 \( 1 + (-0.692 - 0.721i)T \)
53 \( 1 + (0.0402 + 0.999i)T \)
59 \( 1 + (-0.948 + 0.316i)T \)
61 \( 1 + (0.970 + 0.239i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.885 - 0.464i)T \)
73 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (0.948 + 0.316i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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