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Properties

Label 1-297-297.137-r1-0-0
Degree $1$
Conductor $297$
Sign $-0.828 - 0.559i$
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  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (0.719 − 0.694i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (−0.5 + 0.866i)10-s + (0.559 − 0.829i)13-s + (0.615 + 0.788i)14-s + (0.438 − 0.898i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.241 − 0.970i)20-s + (−0.766 − 0.642i)23-s + (0.0348 − 0.999i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (0.848 − 0.529i)4-s + (0.719 − 0.694i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (−0.5 + 0.866i)10-s + (0.559 − 0.829i)13-s + (0.615 + 0.788i)14-s + (0.438 − 0.898i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.241 − 0.970i)20-s + (−0.766 − 0.642i)23-s + (0.0348 − 0.999i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.828 - 0.559i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ -0.828 - 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2503678461 - 0.8177718605i\)
\(L(\frac12)\) \(\approx\) \(0.2503678461 - 0.8177718605i\)
\(L(1)\) \(\approx\) \(0.6826578041 - 0.2274301354i\)
\(L(1)\) \(\approx\) \(0.6826578041 - 0.2274301354i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.961 + 0.275i)T \)
5 \( 1 + (0.719 - 0.694i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
13 \( 1 + (0.559 - 0.829i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.615 - 0.788i)T \)
31 \( 1 + (-0.997 - 0.0697i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.848 - 0.529i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.882 + 0.469i)T \)
61 \( 1 + (-0.997 + 0.0697i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (-0.559 - 0.829i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.719 - 0.694i)T \)
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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

−25.63497017457148460921726270992, −25.0578557552881714400212458528, −24.01886153413662770871338262840, −22.52234367234637043883916809788, −21.78639069649116183702458685667, −21.09241057052991624531551524653, −19.96773379288698009584699055176, −18.99184994998761775362060615624, −18.22639474800325285296311695226, −17.780801389336760307829410210738, −16.39965734682477501780384185244, −15.78210080262197306461013802827, −14.60591173902054559687439193580, −13.50258002577485842096285539801, −12.29122491469354589378976860768, −11.396535279962752507446163806812, −10.47743529394286321452527280279, −9.39752584514155811533969715452, −8.91437676138931777746798053126, −7.4814341709225935616402032622, −6.49342603206189642860929999510, −5.66967119307600659693099685424, −3.65037835845101771757099399456, −2.51408307361474410831387721518, −1.63472024570804977943490676692, 0.352694657319210031769258404745, 1.367031714289391399453591243587, 2.76588377928609610074368090881, 4.4521695808222472006433884651, 5.813581199262085579725868725, 6.61757846831458999339846736769, 7.83359232868605891573143213466, 8.733013216053249037217981598041, 9.74953601193549215422961122748, 10.432127777399665935992699337, 11.47523189570330811876993999145, 12.922204908217279542782033547105, 13.6274591536518793891364199833, 14.88794198212843389546406154771, 16.06842168855906518548135997935, 16.58294313169718762733531991544, 17.73029926488185287275886452440, 18.00589608117001893744701273788, 19.59031720135383537786124176008, 20.120845349758834183548463645039, 20.850604375422845131860293942350, 22.06964310412968228056934637215, 23.27807965333218069000621165709, 24.22300145180757475425365086058, 24.871724377823233677362325848336

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