Properties

Label 1-297-297.281-r0-0-0
Degree 11
Conductor 297297
Sign 0.8280.560i0.828 - 0.560i
Analytic cond. 1.379261.37926
Root an. cond. 1.379261.37926
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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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

Λ(s)=(297s/2ΓR(s)L(s)=((0.8280.560i)Λ(1s)\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}
Λ(s)=(297s/2ΓR(s)L(s)=((0.8280.560i)Λ(1s)\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: 11
Conductor: 297297    =    33113^{3} \cdot 11
Sign: 0.8280.560i0.828 - 0.560i
Analytic conductor: 1.379261.37926
Root analytic conductor: 1.379261.37926
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ297(281,)\chi_{297} (281, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 297, (0: ), 0.8280.560i)(1,\ 297,\ (0:\ ),\ 0.828 - 0.560i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.93225716550.2856748913i0.9322571655 - 0.2856748913i
L(12)L(\frac12) \approx 0.93225716550.2856748913i0.9322571655 - 0.2856748913i
L(1)L(1) \approx 0.84127606210.1871520907i0.8412760621 - 0.1871520907i
L(1)L(1) \approx 0.84127606210.1871520907i0.8412760621 - 0.1871520907i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
11 1 1
good2 1+(0.7190.694i)T 1 + (-0.719 - 0.694i)T
5 1+(0.241+0.970i)T 1 + (0.241 + 0.970i)T
7 1+(0.6150.788i)T 1 + (0.615 - 0.788i)T
13 1+(0.4380.898i)T 1 + (-0.438 - 0.898i)T
17 1+(0.9130.406i)T 1 + (0.913 - 0.406i)T
19 1+(0.669+0.743i)T 1 + (-0.669 + 0.743i)T
23 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
29 1+(0.990+0.139i)T 1 + (0.990 + 0.139i)T
31 1+(0.5590.829i)T 1 + (0.559 - 0.829i)T
37 1+(0.669+0.743i)T 1 + (0.669 + 0.743i)T
41 1+(0.9900.139i)T 1 + (0.990 - 0.139i)T
43 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
47 1+(0.0348+0.999i)T 1 + (-0.0348 + 0.999i)T
53 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
59 1+(0.848+0.529i)T 1 + (-0.848 + 0.529i)T
61 1+(0.5590.829i)T 1 + (-0.559 - 0.829i)T
67 1+(0.1730.984i)T 1 + (0.173 - 0.984i)T
71 1+(0.913+0.406i)T 1 + (-0.913 + 0.406i)T
73 1+(0.9780.207i)T 1 + (0.978 - 0.207i)T
79 1+(0.719+0.694i)T 1 + (0.719 + 0.694i)T
83 1+(0.4380.898i)T 1 + (0.438 - 0.898i)T
89 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
97 1+(0.241+0.970i)T 1 + (-0.241 + 0.970i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line