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Properties

Label 1-297-297.281-r0-0-0
Degree $1$
Conductor $297$
Sign $0.828 - 0.560i$
Analytic cond. $1.37926$
Root an. cond. $1.37926$
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  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.241 + 0.970i)5-s + (0.615 − 0.788i)7-s + (0.669 − 0.743i)8-s + (0.5 − 0.866i)10-s + (−0.438 − 0.898i)13-s + (−0.990 + 0.139i)14-s + (−0.997 + 0.0697i)16-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (−0.961 + 0.275i)20-s + (0.939 − 0.342i)23-s + (−0.882 + 0.469i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.241 + 0.970i)5-s + (0.615 − 0.788i)7-s + (0.669 − 0.743i)8-s + (0.5 − 0.866i)10-s + (−0.438 − 0.898i)13-s + (−0.990 + 0.139i)14-s + (−0.997 + 0.0697i)16-s + (0.913 − 0.406i)17-s + (−0.669 + 0.743i)19-s + (−0.961 + 0.275i)20-s + (0.939 − 0.342i)23-s + (−0.882 + 0.469i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.828 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9322571655 - 0.2856748913i\)
\(L(\frac12)\) \(\approx\) \(0.9322571655 - 0.2856748913i\)
\(L(1)\) \(\approx\) \(0.8412760621 - 0.1871520907i\)
\(L(1)\) \(\approx\) \(0.8412760621 - 0.1871520907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
5 \( 1 + (0.241 + 0.970i)T \)
7 \( 1 + (0.615 - 0.788i)T \)
13 \( 1 + (-0.438 - 0.898i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
31 \( 1 + (0.559 - 0.829i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.990 - 0.139i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.0348 + 0.999i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (-0.559 - 0.829i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (0.719 + 0.694i)T \)
83 \( 1 + (0.438 - 0.898i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.241 + 0.970i)T \)
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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

−25.20728289326402455200025623966, −24.870532006627376344932392764448, −23.853087572040120573610520563667, −23.29975998649956191433222649617, −21.5858952685222655363936114967, −21.13014033599953625054437046065, −19.77537017932262927673023368851, −19.12330052173328964739989595321, −18.03417823722034948621553403334, −17.21747367767355361564304984302, −16.53983644976030508063397226874, −15.5229753751894470647056242962, −14.67813816544697138378075513697, −13.7123020390355927987658119091, −12.42540091563479663296693881505, −11.481797941217888724852525170241, −10.22579588965766746400438806153, −9.10157693868066694723606588209, −8.64819779947675809207446785121, −7.56571235231214379639768316890, −6.3190060068991040011749838566, −5.27849171201231072732389774838, −4.545153612408934811008592172553, −2.300150260486458924617535444241, −1.17509968267510206327350712590, 1.063882247922954773191676500917, 2.49676685849325928365549591997, 3.4143953226431046260472805062, 4.71105263001272074092344434051, 6.37485372391784877197530334376, 7.53998164979235187555600221519, 8.10087300239479282283303833222, 9.651018193439537212264361878481, 10.38515358413118030062406454483, 11.02942217388390378773380807163, 12.0878028773230256679286724012, 13.19820840040140021335009034478, 14.23503639239979014598690192112, 15.1322451857645809242936825805, 16.59045498584676376152297197582, 17.32944660717615330231320497267, 18.13073634404845958423661361353, 18.935511419006281672984990948425, 19.83086672823258861178282662257, 20.84040780494387620980856635884, 21.42411699933484185684619877833, 22.6390897311062395081305105662, 23.17111267657824639898824780290, 24.74444892805627159335047315435, 25.52763714077386424576618261039

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