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Properties

Label 1-79-79.45-r0-0-0
Degree $1$
Conductor $79$
Sign $0.911 - 0.411i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.200 + 0.979i)2-s + (0.428 − 0.903i)3-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (0.799 + 0.600i)6-s + (−0.996 + 0.0804i)7-s + (0.568 − 0.822i)8-s + (−0.632 − 0.774i)9-s + (0.568 + 0.822i)10-s + (0.278 − 0.960i)11-s + (−0.748 + 0.663i)12-s + (0.799 − 0.600i)13-s + (0.120 − 0.992i)14-s + (−0.354 − 0.935i)15-s + (0.692 + 0.721i)16-s + (0.120 + 0.992i)17-s + ⋯
L(s)  = 1  + (−0.200 + 0.979i)2-s + (0.428 − 0.903i)3-s + (−0.919 − 0.391i)4-s + (0.692 − 0.721i)5-s + (0.799 + 0.600i)6-s + (−0.996 + 0.0804i)7-s + (0.568 − 0.822i)8-s + (−0.632 − 0.774i)9-s + (0.568 + 0.822i)10-s + (0.278 − 0.960i)11-s + (−0.748 + 0.663i)12-s + (0.799 − 0.600i)13-s + (0.120 − 0.992i)14-s + (−0.354 − 0.935i)15-s + (0.692 + 0.721i)16-s + (0.120 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8985191551 - 0.1934067593i\)
\(L(\frac12)\) \(\approx\) \(0.8985191551 - 0.1934067593i\)
\(L(1)\) \(\approx\) \(0.9888369039 - 0.04019352493i\)
\(L(1)\) \(\approx\) \(0.9888369039 - 0.04019352493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.200 + 0.979i)T \)
3 \( 1 + (0.428 - 0.903i)T \)
5 \( 1 + (0.692 - 0.721i)T \)
7 \( 1 + (-0.996 + 0.0804i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
17 \( 1 + (0.120 + 0.992i)T \)
19 \( 1 + (-0.0402 + 0.999i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.987 + 0.160i)T \)
31 \( 1 + (0.948 - 0.316i)T \)
37 \( 1 + (-0.845 - 0.534i)T \)
41 \( 1 + (-0.970 - 0.239i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (0.428 + 0.903i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (0.885 - 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (0.568 - 0.822i)T \)
73 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (-0.919 - 0.391i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
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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

−30.99438610313363174076329512942, −30.21885157731654619457906711942, −28.889942608383377226691418122, −28.22290329092510786665687201676, −26.88532905098880463137731412416, −26.03256731773361812879963398270, −25.43024175681355263972136823467, −22.93441244526842182201361561044, −22.367925819403552188396054978060, −21.35413094935212573744462092661, −20.39126452710479261859568047557, −19.37708981878858719430471747114, −18.245183953033948367919831754049, −17.00424246928052989086945292204, −15.655420617980936658471188126180, −14.162532637083136623427916412481, −13.41337638560634323349843127653, −11.76632299611806230994990809913, −10.39615565445245863152680990622, −9.76786369796449817991999401986, −8.77870160001807038180552955172, −6.733941572386944568143054019608, −4.787715399687374150572358023552, −3.401750446444097233695589963289, −2.33407112088409434474212525707, 1.226624025021492206189500384276, 3.54575105551082722921345859903, 5.83872161661476866913850670577, 6.309176815393713613679829907320, 8.07118354409070148079344081224, 8.83547022306482839193290583562, 10.08342208197416547068723870469, 12.44019955496391256870266037537, 13.361032190480821967544022470450, 14.08123100994065920636079620252, 15.6678162786715694054694061372, 16.75376717623091649286239710293, 17.752607941535417622628693949307, 18.891272962532209746134993759881, 19.76248874221028550379545835484, 21.36725863370148439326971485370, 22.80702345203467561518278459741, 23.82951771561567853359268086465, 24.825038196974364003331148458940, 25.48298010105833294554334229869, 26.2991921018557667012195523919, 27.86600617882507551483256071036, 28.94065183207138729262005168199, 29.901566385046366531563779194212, 31.43023731617013648449493726657

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