Properties

Label 1-79-79.16-r0-0-0
Degree $1$
Conductor $79$
Sign $0.365 - 0.930i$
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  + (0.428 − 0.903i)2-s + (0.278 + 0.960i)3-s + (−0.632 − 0.774i)4-s + (−0.200 − 0.979i)5-s + (0.987 + 0.160i)6-s + (0.692 − 0.721i)7-s + (−0.970 + 0.239i)8-s + (−0.845 + 0.534i)9-s + (−0.970 − 0.239i)10-s + (0.948 − 0.316i)11-s + (0.568 − 0.822i)12-s + (0.987 − 0.160i)13-s + (−0.354 − 0.935i)14-s + (0.885 − 0.464i)15-s + (−0.200 + 0.979i)16-s + (−0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (0.428 − 0.903i)2-s + (0.278 + 0.960i)3-s + (−0.632 − 0.774i)4-s + (−0.200 − 0.979i)5-s + (0.987 + 0.160i)6-s + (0.692 − 0.721i)7-s + (−0.970 + 0.239i)8-s + (−0.845 + 0.534i)9-s + (−0.970 − 0.239i)10-s + (0.948 − 0.316i)11-s + (0.568 − 0.822i)12-s + (0.987 − 0.160i)13-s + (−0.354 − 0.935i)14-s + (0.885 − 0.464i)15-s + (−0.200 + 0.979i)16-s + (−0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $0.365 - 0.930i$
Analytic conductor: \(0.366874\)
Root analytic conductor: \(0.366874\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (0:\ ),\ 0.365 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9985221449 - 0.6805939198i\)
\(L(\frac12)\) \(\approx\) \(0.9985221449 - 0.6805939198i\)
\(L(1)\) \(\approx\) \(1.155536575 - 0.5161737955i\)
\(L(1)\) \(\approx\) \(1.155536575 - 0.5161737955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (0.428 - 0.903i)T \)
3 \( 1 + (0.278 + 0.960i)T \)
5 \( 1 + (-0.200 - 0.979i)T \)
7 \( 1 + (0.692 - 0.721i)T \)
11 \( 1 + (0.948 - 0.316i)T \)
13 \( 1 + (0.987 - 0.160i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.919 - 0.391i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.0402 + 0.999i)T \)
31 \( 1 + (-0.996 + 0.0804i)T \)
37 \( 1 + (0.799 - 0.600i)T \)
41 \( 1 + (-0.748 + 0.663i)T \)
43 \( 1 + (0.948 + 0.316i)T \)
47 \( 1 + (0.799 + 0.600i)T \)
53 \( 1 + (0.278 - 0.960i)T \)
59 \( 1 + (-0.632 + 0.774i)T \)
61 \( 1 + (0.120 + 0.992i)T \)
67 \( 1 + (0.568 - 0.822i)T \)
71 \( 1 + (-0.970 + 0.239i)T \)
73 \( 1 + (0.987 + 0.160i)T \)
83 \( 1 + (-0.632 - 0.774i)T \)
89 \( 1 + (-0.970 - 0.239i)T \)
97 \( 1 + (0.120 + 0.992i)T \)
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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

−31.010661249636991494406836702816, −30.69922947862564745448509841564, −29.70047547174976400924584690206, −27.92664120038600966926657761263, −26.77850460680619484098184465466, −25.5631070420437702080691680988, −25.003617930938086324162305054553, −23.867108607496133633004573032750, −22.9211660379064958485516461982, −21.98593543630660475701533806688, −20.52040069993116204122533324157, −18.76497794086454017545143518510, −18.262474943573874862046495141845, −17.13481990978080079290099439409, −15.45184694379914272391160828995, −14.53256268865336774503136862463, −13.79920778596577212295147010197, −12.31964196066580460340474103888, −11.36902405267980066282436149659, −9.0250011229233745716134381665, −7.99303926027430427212461531282, −6.799110589266724715025036492997, −5.96233576696638057913143880774, −3.98334385269458052841270507951, −2.348930268429702039343610851637, 1.51486501856854438484367599209, 3.75321556594078605837900223806, 4.35841259273153582480314574693, 5.73969366648744700734820966748, 8.394596457714266308883253201399, 9.204043431975541802725128674923, 10.68222131346242107090693844079, 11.43025022550223699970309161223, 12.98264118591329940936444929001, 14.061683462147946902063933870421, 15.13303029071679145854382488877, 16.529749438520316658313533902724, 17.6373297713395995939697485858, 19.59203179683089557262017824716, 20.11728703743084686525895543948, 21.135353377948536251426421871156, 21.85425055382387360807486873381, 23.309343055708213366840667300219, 24.117333039761272475435687470813, 25.70657851880500963219714342472, 27.27159177948096428946978751395, 27.658759960343781156469389446292, 28.61926909255737433884799240098, 30.053154980348358218123510728558, 30.93566637164924216439673059189

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