↑

Properties

Label 1-79-79.71-r1-0-0
Degree $1$
Conductor $79$
Sign $0.942 + 0.334i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.748 − 0.663i)2-s + (−0.568 − 0.822i)3-s + (0.120 + 0.992i)4-s + (−0.970 − 0.239i)5-s + (−0.120 + 0.992i)6-s + (−0.568 − 0.822i)7-s + (0.568 − 0.822i)8-s + (−0.354 + 0.935i)9-s + (0.568 + 0.822i)10-s + (−0.970 + 0.239i)11-s + (0.748 − 0.663i)12-s + (0.120 + 0.992i)13-s + (−0.120 + 0.992i)14-s + (0.354 + 0.935i)15-s + (−0.970 + 0.239i)16-s + (−0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (−0.748 − 0.663i)2-s + (−0.568 − 0.822i)3-s + (0.120 + 0.992i)4-s + (−0.970 − 0.239i)5-s + (−0.120 + 0.992i)6-s + (−0.568 − 0.822i)7-s + (0.568 − 0.822i)8-s + (−0.354 + 0.935i)9-s + (0.568 + 0.822i)10-s + (−0.970 + 0.239i)11-s + (0.748 − 0.663i)12-s + (0.120 + 0.992i)13-s + (−0.120 + 0.992i)14-s + (0.354 + 0.935i)15-s + (−0.970 + 0.239i)16-s + (−0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ 0.942 + 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2820110748 + 0.04856175788i\)
\(L(\frac12)\) \(\approx\) \(0.2820110748 + 0.04856175788i\)
\(L(1)\) \(\approx\) \(0.3805679131 - 0.1961946668i\)
\(L(1)\) \(\approx\) \(0.3805679131 - 0.1961946668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.748 - 0.663i)T \)
3 \( 1 + (-0.568 - 0.822i)T \)
5 \( 1 + (-0.970 - 0.239i)T \)
7 \( 1 + (-0.568 - 0.822i)T \)
11 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + (0.120 + 0.992i)T \)
17 \( 1 + (-0.120 - 0.992i)T \)
19 \( 1 + (0.885 - 0.464i)T \)
23 \( 1 + T \)
29 \( 1 + (0.354 + 0.935i)T \)
31 \( 1 + (-0.748 - 0.663i)T \)
37 \( 1 + (-0.885 + 0.464i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (0.970 + 0.239i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (-0.568 + 0.822i)T \)
59 \( 1 + (-0.120 + 0.992i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.13493596517722685131129298407, −29.18694979347523701179719103411, −28.381990380598412794358268386020, −27.514743035318708089775668151641, −26.64899781527363212275821899723, −25.76753654943630147440166313323, −24.37646971148439243259893732131, −23.17101885045431980136921227040, −22.53607004934361255953080848775, −20.96312116674077391536460697752, −19.62718898531657762314804200818, −18.58898000995772709015303720362, −17.52487302039007679515297440939, −16.0415314330438218328142699140, −15.68416759939970551670660746357, −14.74775312496614098471195918899, −12.5699402928291675758114149049, −11.11611107710074318901691123636, −10.266331977734274393969129404044, −8.91409277339960674175917411589, −7.75115269635902619607616855841, −6.14222429795289107897934507238, −5.12207485306837183711973524525, −3.223691371685253413153250890557, −0.244835603723904535812164964523, 1.00294028475315768430039062144, 2.95760829295619675212927762648, 4.671624900795304244199913190391, 7.04392505529466401157211476698, 7.55574200607471852864632905782, 9.118504368783636012737474240147, 10.72100643730175558544002699399, 11.571932556573552695492837585879, 12.65705607843698855594804984962, 13.603374934778381951262643720377, 16.01008130694550030941751101388, 16.62812068411011217460798171672, 17.97582303691757541782438299832, 18.910837066852939417763522864917, 19.769676978444262551229022425471, 20.7514581678850903934890529397, 22.44203321937679960856791964052, 23.320091550712195594741110784856, 24.36112524477251815624281238767, 25.85807155045161869790077362138, 26.78338785743670325784899943976, 27.920019579672066454172404320439, 28.960805634176247401096551575889, 29.4675292522385376657415986620, 30.94767037100526295622005341596

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