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Properties

Label 1-297-297.38-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  + (0.719 + 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

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

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: $\chi_{297} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ -0.828 - 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5824998027 + 1.902608320i\)
\(L(\frac12)\) \(\approx\) \(-0.5824998027 + 1.902608320i\)
\(L(1)\) \(\approx\) \(0.8701763317 + 1.065322507i\)
\(L(1)\) \(\approx\) \(0.8701763317 + 1.065322507i\)

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

−24.64192294447627230486768924209, −23.46473728537194894334270034275, −22.91245940536854825171838210869, −21.956586565322772731508923744505, −20.85483031471163731072010380916, −20.26527439115191267923195460858, −19.66054045388139425576179415452, −18.44435263567655048194095968299, −17.35954256929732151480990854280, −16.24170877118143737319290729788, −15.488479635789474730825176642277, −14.16557088177191885796375118615, −13.22749251271265077809087659485, −12.87616652356390665542448165814, −11.69412142067448681625449816377, −10.62947387677001763777454373998, −9.74893905237020779570397317231, −8.8188578699878670728829003451, −7.30075380308497265634868001963, −6.020499041490539598626169629775, −5.109210922540733955430175048618, −4.04009254360195226363702674303, −3.03199815880009193116693481439, −1.47436158564015122241487730113, −0.45707494260898525712209614722, 2.2826455010276898086543393087, 3.16383988754347964589908808688, 4.3635360554784234247599585889, 5.72344264240585557784291909810, 6.5029186756632909742728303988, 7.23184026034386235658844441074, 8.66053735892187350144708034165, 9.520275855780952232492774588304, 11.08045775815496206439680115886, 11.79809012484137327696417603175, 13.11552806728770416079642752066, 13.6808821743362563957419412559, 14.90949878729398404202356494462, 15.37550906909228041624056509498, 16.394495158657325340323404458868, 17.43574303823464683066522815785, 18.41131261315949493070937030359, 19.16743959889895336927120539482, 20.59351738976617447055830683046, 21.657448851267845127823073116249, 22.205194527920473097163368985203, 22.86558132211425692924991504807, 23.97323078334814308550739102703, 24.75544931231149480827355229756, 25.8716054352566296576287021311

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