Properties

Label 1-79-79.38-r0-0-0
Degree $1$
Conductor $79$
Sign $-0.999 + 0.0291i$
Analytic cond. $0.366874$
Root an. cond. $0.366874$
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  + (0.568 − 0.822i)2-s + (−0.970 + 0.239i)3-s + (−0.354 − 0.935i)4-s + (−0.748 − 0.663i)5-s + (−0.354 + 0.935i)6-s + (−0.970 + 0.239i)7-s + (−0.970 − 0.239i)8-s + (0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (−0.748 + 0.663i)11-s + (0.568 + 0.822i)12-s + (−0.354 − 0.935i)13-s + (−0.354 + 0.935i)14-s + (0.885 + 0.464i)15-s + (−0.748 + 0.663i)16-s + (−0.354 − 0.935i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (−0.970 + 0.239i)3-s + (−0.354 − 0.935i)4-s + (−0.748 − 0.663i)5-s + (−0.354 + 0.935i)6-s + (−0.970 + 0.239i)7-s + (−0.970 − 0.239i)8-s + (0.885 − 0.464i)9-s + (−0.970 + 0.239i)10-s + (−0.748 + 0.663i)11-s + (0.568 + 0.822i)12-s + (−0.354 − 0.935i)13-s + (−0.354 + 0.935i)14-s + (0.885 + 0.464i)15-s + (−0.748 + 0.663i)16-s + (−0.354 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.999 + 0.0291i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ -0.999 + 0.0291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006712663515 - 0.4597164162i\)
\(L(\frac12)\) \(\approx\) \(0.006712663515 - 0.4597164162i\)
\(L(1)\) \(\approx\) \(0.4956688639 - 0.4371763492i\)
\(L(1)\) \(\approx\) \(0.4956688639 - 0.4371763492i\)

Euler product

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

−31.68561122363597179015568782938, −30.74722370352253828620683185941, −29.5960978314900031728277945892, −28.66567098817077649057178406448, −26.94008872667661497896950585852, −26.490749746361370986867336766632, −25.03580014601100804101789386663, −23.66845931640718436025299795674, −23.30544167801477509089562372719, −22.25663949295991109513404140778, −21.393242282980183723228759038703, −19.27433933011644983510788322382, −18.43867687161696699655302997695, −16.96677589352864290445341475502, −16.19673900947591170343799709743, −15.24787069994502578084811533909, −13.7341428991814574963705302745, −12.62412879325256249440897070432, −11.58044353943195508499653363515, −10.22704106290584051467474145094, −8.18044042325258519006991173802, −6.89008765155185513092859697507, −6.20772552776251373841139231051, −4.61370217739925500668544295811, −3.23760920000216513767804819629, 0.4726477665109222057669849304, 2.96023915517940303071469908056, 4.57432774026279436861700315948, 5.388740592204116219794686792797, 7.03188952267878918168060333275, 9.20469400440928735373643939747, 10.30201169655564304077148610902, 11.501180009455809397406243155217, 12.54245482958539212404443799807, 13.12702541050627472363481927750, 15.37026494217285653324062127754, 15.829005543592686208774763713064, 17.467249080896269094352459627030, 18.70406261216492000096591084858, 19.89318331183596206086322055286, 20.80966263083816636260846138006, 22.15532217440994956572738838277, 22.90218971022692629680709505887, 23.64878622017571092529111411818, 24.88225348977305468585275443579, 26.8194791581694592641871694172, 27.80128280556002968312733397136, 28.64157261223168350547236794882, 29.25157025733688256397247402308, 30.58741724582997708975213042465

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