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Properties

Label 1-297-297.139-r1-0-0
Degree $1$
Conductor $297$
Sign $0.999 + 0.0235i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)5-s + (−0.438 − 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.5 + 0.866i)10-s + (−0.848 − 0.529i)13-s + (−0.997 + 0.0697i)14-s + (0.0348 + 0.999i)16-s + (0.978 − 0.207i)17-s + (−0.913 + 0.406i)19-s + (0.990 − 0.139i)20-s + (0.173 + 0.984i)23-s + (−0.241 − 0.970i)25-s + (−0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)5-s + (−0.438 − 0.898i)7-s + (−0.913 + 0.406i)8-s + (0.5 + 0.866i)10-s + (−0.848 − 0.529i)13-s + (−0.997 + 0.0697i)14-s + (0.0348 + 0.999i)16-s + (0.978 − 0.207i)17-s + (−0.913 + 0.406i)19-s + (0.990 − 0.139i)20-s + (0.173 + 0.984i)23-s + (−0.241 − 0.970i)25-s + (−0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.999 + 0.0235i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.999 + 0.0235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.108072537 + 0.01304853622i\)
\(L(\frac12)\) \(\approx\) \(1.108072537 + 0.01304853622i\)
\(L(1)\) \(\approx\) \(0.8452901099 - 0.3470244499i\)
\(L(1)\) \(\approx\) \(0.8452901099 - 0.3470244499i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
5 \( 1 + (-0.615 + 0.788i)T \)
7 \( 1 + (-0.438 - 0.898i)T \)
13 \( 1 + (-0.848 - 0.529i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.997 + 0.0697i)T \)
31 \( 1 + (-0.882 + 0.469i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (0.997 - 0.0697i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.719 + 0.694i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (0.882 + 0.469i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−25.04208190936751366120425918051, −24.30600778542685762141288888627, −23.53949189427018829077001504538, −22.67661258527049657922215278635, −21.67727003959375693276126688322, −21.000345326608760472622690183163, −19.55366953383034368815877365952, −18.8621507048727738537207216713, −17.67405624024172920524595805284, −16.58239269385790875136836319298, −16.178757759799257223942782203908, −15.04175867628418346668901652500, −14.46419566644899130721774734813, −12.908059796490101527109065760045, −12.52867801130910598885347958396, −11.57218504358057992990885359609, −9.7497996462150064009947179577, −8.85675740715613638961741932573, −8.087129766815751647764499079260, −6.9676287244380706291511472817, −5.86371770644088503748721975744, −4.886412822986081756280421355956, −3.965641654034544804599959220089, −2.55915481920188186667193041, −0.382834677616864245425270406213, 0.92303509318712725631920927982, 2.62028151058386768291537731153, 3.50248704800667462344483841317, 4.40895160445655698061087313753, 5.76723827528565672485493045738, 7.04626244023824728293758191322, 8.03250426893427029981481392451, 9.61155896748122371618240043619, 10.346444042449925316546445236252, 11.104166637330289616372234298077, 12.18485230481046599001745616412, 12.96813242664617046717889138413, 14.18123236312591223454431270830, 14.71173364922515294612952530633, 15.88142612580241670927776271612, 17.16603064517229604283896121192, 18.14462710098420584013241752042, 19.31373205008996166581170476404, 19.55727544574220025388693675927, 20.626909431159706262012906525449, 21.65184204091045419651054485735, 22.51026271446058951645811533737, 23.29235739441394503434335437954, 23.72106282082097843301913113407, 25.26644217670191984892813521796

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