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Properties

Label 1-279-279.11-r0-0-0
Degree $1$
Conductor $279$
Sign $0.992 + 0.119i$
Analytic cond. $1.29567$
Root an. cond. $1.29567$
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  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (0.5 + 0.866i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)10-s + (0.309 − 0.951i)11-s + (0.104 − 0.994i)13-s + (−0.309 − 0.951i)14-s + (−0.104 + 0.994i)16-s + (0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (−0.309 + 0.951i)20-s + (−0.669 + 0.743i)22-s + (−0.978 − 0.207i)23-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.669 + 0.743i)4-s + (0.5 + 0.866i)5-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)10-s + (0.309 − 0.951i)11-s + (0.104 − 0.994i)13-s + (−0.309 − 0.951i)14-s + (−0.104 + 0.994i)16-s + (0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (−0.309 + 0.951i)20-s + (−0.669 + 0.743i)22-s + (−0.978 − 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $0.992 + 0.119i$
Analytic conductor: \(1.29567\)
Root analytic conductor: \(1.29567\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 279,\ (0:\ ),\ 0.992 + 0.119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004689835 + 0.06017495373i\)
\(L(\frac12)\) \(\approx\) \(1.004689835 + 0.06017495373i\)
\(L(1)\) \(\approx\) \(0.8808413060 + 0.0003681582961i\)
\(L(1)\) \(\approx\) \(0.8808413060 + 0.0003681582961i\)

Euler product

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

−25.64318173213582911822296901282, −24.83239732690051673874855692189, −23.88982585319748353464065947590, −23.447477784658419488802480102064, −21.782716566738502264979964209972, −20.71988713630886190694999331922, −20.18573671971243113343903839209, −19.24180235828963697502940237430, −17.92622537479668131139772382448, −17.40341563398125224930355005777, −16.622445741951133232588787503268, −15.75381265806246754997359841778, −14.45302770495333002809135668745, −13.826255543513689234301004821434, −12.32175166286263088021551338819, −11.40497564059268560602930378646, −10.11608962747017093623907441592, −9.51175273900376970635844421559, −8.40248415480721031211954321795, −7.52431317107827906879682329295, −6.44607493157323764347028964053, −5.23689580326871716450518530398, −4.20237240980689121535451306609, −2.001550553891523506468638797468, −1.18912323791102214964470253585, 1.272687749702710720946516042670, 2.63563668984843156558706080661, 3.38494540992930390541633495121, 5.42659342271567785314942702205, 6.39454622793954860122120279096, 7.69683155283862019959316469779, 8.45764554018202265797799681996, 9.65335114746919260741558044594, 10.44322449483086965293343021739, 11.45330505712543794036823561334, 12.11717577944825413602418909172, 13.58735042298900233932403020493, 14.560068594939465874904982752650, 15.64430021936423024715279291474, 16.58937630675739423731683226623, 17.842022008754179567217074941635, 18.23465654839128176484657239470, 19.00076550154595827951920302184, 20.156794270784348064676170205971, 21.0764218742493501612867110216, 21.879730129069790500626769284140, 22.56307460249584692410473965324, 24.16841189229760110769876872148, 25.10984955604376973508268276289, 25.56347061396284419091734328059

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