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Properties

Label 1-279-279.166-r1-0-0
Degree $1$
Conductor $279$
Sign $0.451 - 0.892i$
Analytic cond. $29.9827$
Root an. cond. $29.9827$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + 5-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)10-s + (0.978 − 0.207i)11-s + (0.809 + 0.587i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)17-s + (0.913 + 0.406i)19-s + (−0.978 + 0.207i)20-s + (−0.309 − 0.951i)22-s + (−0.669 + 0.743i)23-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + 5-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)10-s + (0.978 − 0.207i)11-s + (0.809 + 0.587i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)17-s + (0.913 + 0.406i)19-s + (−0.978 + 0.207i)20-s + (−0.309 − 0.951i)22-s + (−0.669 + 0.743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $0.451 - 0.892i$
Analytic conductor: \(29.9827\)
Root analytic conductor: \(29.9827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 279,\ (1:\ ),\ 0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.992115545 - 1.224236086i\)
\(L(\frac12)\) \(\approx\) \(1.992115545 - 1.224236086i\)
\(L(1)\) \(\approx\) \(1.175540182 - 0.5768271371i\)
\(L(1)\) \(\approx\) \(1.175540182 - 0.5768271371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−25.30674610964466403506762680585, −24.83097092646825752982445176053, −24.168996126208210657352636792739, −22.590086886831394496178284444744, −22.32650995687954917156543646557, −21.2416820761517921016189212424, −20.12079174807293317949654988158, −18.73859384234261113774521843835, −17.9615194620617664760223206441, −17.48934801616337572482200228067, −16.26405398858261781693661196199, −15.46661944312658628382831218112, −14.43974361226370948467886508220, −13.73711001772440160847001842043, −12.74786067237765886047166593136, −11.50068072839060527240241769119, −10.055434367071846124951060405951, −9.157778008340126931645529131, −8.519643673554559285122849880185, −7.11834943673628326590140566288, −6.07823690946372511484352039822, −5.44865431731674406964942546750, −4.21155556724587028178835961983, −2.49082124815997082980203913919, −0.97806292491941049272599613420, 1.129290259658032502154303475963, 1.800028289841250387799474292607, 3.44154847829336035712628979387, 4.30735243269770153404957642276, 5.64635463451850802669080379936, 6.85496791428562894605295184048, 8.35790811414084315418158601610, 9.27999125036922144133231055607, 10.175957374015792698414519484919, 11.037540551341715160779372994131, 11.94219967673010707622764777142, 13.28249004207189168880796623164, 13.80520107501598912302255690030, 14.55669766454358784028622245880, 16.40210122581902405253316708893, 17.24714130564202805249360751848, 17.9230395577735393331987541939, 18.89851558556267089389100618814, 20.06154586594466412470511668197, 20.53926022048025501323395763186, 21.69751470370046641050230297257, 22.11288841895989876687607449083, 23.338423660946926659784812865227, 24.17991164271344992094879686610, 25.481237109753583433370227303086

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