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Properties

Label 1-79-79.24-r1-0-0
Degree $1$
Conductor $79$
Sign $-0.660 + 0.750i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + 10-s + (−0.5 + 0.866i)11-s − 12-s + (−0.5 − 0.866i)13-s − 14-s − 15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + 10-s + (−0.5 + 0.866i)11-s − 12-s + (−0.5 − 0.866i)13-s − 14-s − 15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(79\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(8.48972\)
Root analytic conductor: \(8.48972\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{79} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 79,\ (1:\ ),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2591312027 + 0.5729881706i\)
\(L(\frac12)\) \(\approx\) \(0.2591312027 + 0.5729881706i\)
\(L(1)\) \(\approx\) \(0.6931354351 + 0.1388887096i\)
\(L(1)\) \(\approx\) \(0.6931354351 + 0.1388887096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad79 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−30.88456813554126898036921975740, −29.05517974589062458457829576095, −28.36183717618534361476933921529, −27.08034424591647756469449486387, −26.09600174297423202782468235545, −24.87206323042271199908413258211, −24.12585589197503258520168525977, −23.739505241899892146917854849498, −21.8619217339536107681364997159, −20.27897087218842664790567053308, −19.23591816282825442393284148248, −18.42763840040317848507000426371, −17.2858509173754003253668408186, −16.03711654615577450962069137638, −14.98261444515685794361045582098, −13.80238736711178171289244435720, −12.624099679762209043731956284, −11.25836445486919501719680259341, −8.9826821350950243039836851533, −8.62264154285051722398373688307, −7.41076780881124883103795742200, −6.02761394964115363707221833242, −4.607189550204918181256467234402, −2.09589542090420501131772349408, −0.32320963911797269691310969312, 2.28114759555439074710595778466, 3.62138650610622922078562051413, 4.65931330172815543677484665502, 7.41518040024451648459398997378, 8.26239063822871945943705273571, 10.03514193157784654518694936799, 10.47675528369271034102515231360, 11.65158341676752793576416263493, 13.3134124731165924415395260290, 14.56562476992785457132482568409, 15.65223698416750872959632290687, 17.16680694067640466984517397570, 18.14584981345222854045380986294, 19.65443494997227104086037022334, 20.17044759663811385931795448616, 21.28118067298216347707523698219, 22.3897349994655473396041977455, 23.25743575967097478340717416180, 25.36296560573469226645139518398, 26.32580117409090072060859484067, 27.11024262112550356442490832928, 27.68564258309021010181909581466, 29.14336969534140401856709403024, 30.36950155286839596223439169901, 30.96252076928684110155256230744

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