Properties

Label 1-79-79.66-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.768 - 0.639i$
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  + (−0.0402 − 0.999i)2-s + (0.919 − 0.391i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.428 − 0.903i)6-s + (−0.799 − 0.600i)7-s + (0.120 + 0.992i)8-s + (0.692 − 0.721i)9-s + (0.120 − 0.992i)10-s + (−0.632 − 0.774i)11-s + (−0.885 + 0.464i)12-s + (0.428 − 0.903i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (0.987 − 0.160i)16-s + (−0.568 − 0.822i)17-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + (0.919 − 0.391i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.428 − 0.903i)6-s + (−0.799 − 0.600i)7-s + (0.120 + 0.992i)8-s + (0.692 − 0.721i)9-s + (0.120 − 0.992i)10-s + (−0.632 − 0.774i)11-s + (−0.885 + 0.464i)12-s + (0.428 − 0.903i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (0.987 − 0.160i)16-s + (−0.568 − 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.768 - 0.639i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.768 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6853233399 - 1.894885574i\)
\(L(\frac12)\) \(\approx\) \(0.6853233399 - 1.894885574i\)
\(L(1)\) \(\approx\) \(0.9886460003 - 0.9666116484i\)
\(L(1)\) \(\approx\) \(0.9886460003 - 0.9666116484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.0402 - 0.999i)T \)
3 \( 1 + (0.919 - 0.391i)T \)
5 \( 1 + (0.987 + 0.160i)T \)
7 \( 1 + (-0.799 - 0.600i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (0.428 - 0.903i)T \)
17 \( 1 + (-0.568 - 0.822i)T \)
19 \( 1 + (0.948 - 0.316i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.278 + 0.960i)T \)
31 \( 1 + (-0.845 + 0.534i)T \)
37 \( 1 + (0.200 - 0.979i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (0.632 - 0.774i)T \)
47 \( 1 + (0.200 + 0.979i)T \)
53 \( 1 + (0.919 + 0.391i)T \)
59 \( 1 + (0.996 + 0.0804i)T \)
61 \( 1 + (0.748 + 0.663i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (-0.996 + 0.0804i)T \)
89 \( 1 + (0.120 - 0.992i)T \)
97 \( 1 + (-0.748 - 0.663i)T \)
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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

−31.46473801141163783011976218645, −30.74436407161649183948846545108, −28.83584896544795430700802756688, −28.040860135020550485237718899294, −26.25619540797083622269893079813, −26.0766942741211153751482240874, −25.05444730512915142546339887410, −24.16627781823823991690128859592, −22.51915220136711055612983528541, −21.68910825291827557150066830098, −20.5187281687192293437103281975, −18.97979179895665618367665891766, −18.10713905442003105665913517734, −16.638567031981829814665782792170, −15.72034664384080317581273277728, −14.679525919600931088914829977580, −13.572801290683409362283890485843, −12.77547553349603959156260523615, −10.08520854909369121211771785280, −9.39034524741908150010911049610, −8.359861515614223976894884128210, −6.80945015744712713232044722289, −5.54263066036840103134521821177, −4.06680500102672849920694493409, −2.20550769778592720697942689649, 0.94233912607517186640046904553, 2.6200945908626861382630837774, 3.50389633598728632477832691052, 5.57794743191500333305160159495, 7.380882901438679668669969249635, 8.92354462629879402020250738745, 9.80859999141005567690627703863, 10.90991562661176897992738957769, 12.80372144992506270555992215071, 13.45397635265023830092418878517, 14.17115135073925207556620274937, 16.00944774067622345604047579878, 17.81023498866651242622722373228, 18.436974453575540548358930523157, 19.75616711744365494487930971021, 20.44564112230161728691944872506, 21.53476902881195380999281579363, 22.595614849193329138318806922189, 23.94889058723322442019658807613, 25.3901519671595596508067892510, 26.224200980390483475074126021206, 27.09584799302363765062630216504, 28.81954348249945615701825928721, 29.500919613985075302852930231786, 30.18457629310940447203891877829

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