Properties

Label 1-79-79.77-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.576 - 0.817i$
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  + (0.278 + 0.960i)2-s + (−0.948 − 0.316i)3-s + (−0.845 + 0.534i)4-s + (0.428 + 0.903i)5-s + (0.0402 − 0.999i)6-s + (0.200 + 0.979i)7-s + (−0.748 − 0.663i)8-s + (0.799 + 0.600i)9-s + (−0.748 + 0.663i)10-s + (−0.996 + 0.0804i)11-s + (0.970 − 0.239i)12-s + (−0.0402 − 0.999i)13-s + (−0.885 + 0.464i)14-s + (−0.120 − 0.992i)15-s + (0.428 − 0.903i)16-s + (−0.885 − 0.464i)17-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)2-s + (−0.948 − 0.316i)3-s + (−0.845 + 0.534i)4-s + (0.428 + 0.903i)5-s + (0.0402 − 0.999i)6-s + (0.200 + 0.979i)7-s + (−0.748 − 0.663i)8-s + (0.799 + 0.600i)9-s + (−0.748 + 0.663i)10-s + (−0.996 + 0.0804i)11-s + (0.970 − 0.239i)12-s + (−0.0402 − 0.999i)13-s + (−0.885 + 0.464i)14-s + (−0.120 − 0.992i)15-s + (0.428 − 0.903i)16-s + (−0.885 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.576 - 0.817i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.576 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2087390366 + 0.4028048234i\)
\(L(\frac12)\) \(\approx\) \(-0.2087390366 + 0.4028048234i\)
\(L(1)\) \(\approx\) \(0.5056604298 + 0.4667572946i\)
\(L(1)\) \(\approx\) \(0.5056604298 + 0.4667572946i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.278 + 0.960i)T \)
3 \( 1 + (-0.948 - 0.316i)T \)
5 \( 1 + (0.428 + 0.903i)T \)
7 \( 1 + (0.200 + 0.979i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (-0.0402 - 0.999i)T \)
17 \( 1 + (-0.885 - 0.464i)T \)
19 \( 1 + (-0.632 - 0.774i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.919 + 0.391i)T \)
31 \( 1 + (0.692 - 0.721i)T \)
37 \( 1 + (-0.987 + 0.160i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (0.996 + 0.0804i)T \)
47 \( 1 + (-0.987 - 0.160i)T \)
53 \( 1 + (-0.948 + 0.316i)T \)
59 \( 1 + (0.845 + 0.534i)T \)
61 \( 1 + (0.354 - 0.935i)T \)
67 \( 1 + (-0.970 + 0.239i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (-0.0402 + 0.999i)T \)
83 \( 1 + (-0.845 + 0.534i)T \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−29.87159167715859837180787619716, −28.88425247205193120877023101214, −28.52614106454640154568761494221, −27.2448581514344062405739422443, −26.34492291809314985463682712101, −24.13093813190475277935858093984, −23.64835116416968234700333958514, −22.487903602381370863971634595775, −21.13866799775846554473270728817, −20.85858138961286575479106735392, −19.37503035017835976787533417507, −17.936428201693665036428761314, −17.10015032990833728732127086981, −15.93414281000545144148071881444, −14.101970575251340518561599373090, −13.01471337648301313949776243857, −12.0639240181291894049857320569, −10.71809822946092188885438571058, −10.039100458884774024811797712987, −8.5370234537247067615325944454, −6.32173044877198126612459750853, −4.88837453439643980921624974655, −4.15479468543507171061583199743, −1.74991001607292053345525787866, −0.22055314663802217399080278818, 2.65686925934767699687626992885, 4.93507589894562404898949647228, 5.85578405402496335504431261914, 6.8911586959560736668336323116, 8.15725561638038807105757270538, 9.91537183403233179811599363938, 11.27788705717890500224639737882, 12.68804149053996315933433263300, 13.63470323206760375649650055713, 15.24899945692820702577956589142, 15.78594745773183833476967387396, 17.63305832348557818377620521934, 17.872718728291117289726018047, 18.94832777196384324221754975932, 21.352990831440575697645054340090, 22.12938152773048180732397696581, 22.93189149556275525485390514745, 24.021081780447222869211259616930, 25.03823356172052520639332816417, 25.977484634186315006376091597528, 27.242649874876449008823976434877, 28.25621010534622193413380522621, 29.54280755344933373241301552912, 30.57789145498471059464522415632, 31.58386292160122561862789639127

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