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Properties

Label 1-297-297.277-r1-0-0
Degree $1$
Conductor $297$
Sign $0.736 - 0.676i$
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  + (0.997 − 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.559 − 0.829i)5-s + (0.882 − 0.469i)7-s + (0.978 − 0.207i)8-s + (0.5 − 0.866i)10-s + (0.241 + 0.970i)13-s + (0.848 − 0.529i)14-s + (0.961 − 0.275i)16-s + (0.104 + 0.994i)17-s + (0.978 − 0.207i)19-s + (0.438 − 0.898i)20-s + (0.173 − 0.984i)23-s + (−0.374 − 0.927i)25-s + (0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (0.997 − 0.0697i)2-s + (0.990 − 0.139i)4-s + (0.559 − 0.829i)5-s + (0.882 − 0.469i)7-s + (0.978 − 0.207i)8-s + (0.5 − 0.866i)10-s + (0.241 + 0.970i)13-s + (0.848 − 0.529i)14-s + (0.961 − 0.275i)16-s + (0.104 + 0.994i)17-s + (0.978 − 0.207i)19-s + (0.438 − 0.898i)20-s + (0.173 − 0.984i)23-s + (−0.374 − 0.927i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.736 - 0.676i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.646274106 - 1.811180603i\)
\(L(\frac12)\) \(\approx\) \(4.646274106 - 1.811180603i\)
\(L(1)\) \(\approx\) \(2.453078929 - 0.5436000996i\)
\(L(1)\) \(\approx\) \(2.453078929 - 0.5436000996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.997 - 0.0697i)T \)
5 \( 1 + (0.559 - 0.829i)T \)
7 \( 1 + (0.882 - 0.469i)T \)
13 \( 1 + (0.241 + 0.970i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.848 - 0.529i)T \)
31 \( 1 + (-0.719 + 0.694i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.848 + 0.529i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.990 + 0.139i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.615 - 0.788i)T \)
61 \( 1 + (0.719 + 0.694i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.997 - 0.0697i)T \)
83 \( 1 + (0.241 - 0.970i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.559 + 0.829i)T \)
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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

−25.157282351021985665864104380223, −24.413649938851120595851039092130, −23.39478466386961357875288208224, −22.37361215454305125239585622557, −21.99947986772355882788363674738, −20.820204143540263542517077908329, −20.35928568074673327156787444917, −18.8584301691742781186085014471, −18.017562806844556333325539063515, −17.103623190086485059075437733857, −15.71757579592024892104229096005, −15.09809876391427026197055623794, −14.129608320932175207133374637837, −13.554070319397190005989497207082, −12.30483488478288458908822527349, −11.35055459639149450251443132771, −10.65368190459227464814000565464, −9.38319815102772298230901039148, −7.790321614211966168414268797774, −7.084973666263213494061372681620, −5.61501739373833162974817696836, −5.306773528892711464114152735676, −3.630106170947272262726667927599, −2.692126403146808938350565585914, −1.55034552754743456089875074629, 1.23298076641867537753808166435, 2.075455910265139440499324281939, 3.77139074537581108771968142758, 4.68159076778124277626175266437, 5.52672672370470229520813561863, 6.65193535455308064465586918043, 7.82451104157810815533082328827, 8.98033667149916227428490304052, 10.29550261650863093667305391231, 11.25035902924605132766184967764, 12.20058333970763871441935623219, 13.12808077931342507618889427292, 13.99780579337655771303048571885, 14.62606892755908764858805943211, 15.92179685633448216154718335176, 16.7433033137869667325299956455, 17.523822297945141961155205682525, 18.892259445189449308930623919071, 20.113370459733802900770824825503, 20.730635525597968973074939148374, 21.41326180643301045361780939281, 22.25831676782641771238472160811, 23.51202971790854922983643809419, 24.11455241906353337720337710517, 24.69087957693099140195806059880

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