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Properties

Label 1-297-297.236-r1-0-0
Degree $1$
Conductor $297$
Sign $0.899 - 0.437i$
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  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.961 − 0.275i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.5 + 0.866i)10-s + (−0.997 − 0.0697i)13-s + (0.374 − 0.927i)14-s + (0.559 + 0.829i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.719 + 0.694i)20-s + (−0.173 + 0.984i)23-s + (0.848 + 0.529i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.961 − 0.275i)5-s + (0.990 + 0.139i)7-s + (−0.669 + 0.743i)8-s + (−0.5 + 0.866i)10-s + (−0.997 − 0.0697i)13-s + (0.374 − 0.927i)14-s + (0.559 + 0.829i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (0.719 + 0.694i)20-s + (−0.173 + 0.984i)23-s + (0.848 + 0.529i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.335625929 - 0.3078031813i\)
\(L(\frac12)\) \(\approx\) \(1.335625929 - 0.3078031813i\)
\(L(1)\) \(\approx\) \(0.8781237857 - 0.4019030349i\)
\(L(1)\) \(\approx\) \(0.8781237857 - 0.4019030349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
5 \( 1 + (-0.961 - 0.275i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
13 \( 1 + (-0.997 - 0.0697i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.374 + 0.927i)T \)
31 \( 1 + (0.438 + 0.898i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.374 - 0.927i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.882 - 0.469i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.0348 - 0.999i)T \)
61 \( 1 + (0.438 - 0.898i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.241 + 0.970i)T \)
83 \( 1 + (0.997 - 0.0697i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−24.759264005116578811921072201530, −24.54616333548723121389133844103, −23.53588849312788694914754092101, −22.73346946042209740705200341055, −21.9849782513887529259318056414, −20.81945357912458846920617652220, −19.799491337562996749951494934760, −18.62999922662496109922469506628, −17.907446932621505423449600518700, −16.893127087389659351650283223147, −16.03474142099090869527591155509, −15.00966291244110026335524544101, −14.53483533593032171629307040331, −13.4858508344130818533937577405, −12.19441372245467610470460976774, −11.508059395621370729122809399729, −10.143862401079212719847165679270, −8.83347196573405984650603054945, −7.83615358027153953385265156572, −7.33412949647665577454378590054, −6.08372884992270858990024952353, −4.701620428946553097330071436774, −4.20389853794608722963255908006, −2.648634401622934575740708744993, −0.51068500349713384009459296363, 0.92366151399937303088826481243, 2.23635084668329286372217795807, 3.51224256723368159109750910139, 4.656963906499477974191486364926, 5.23442978642643786392342347610, 7.11215617137232779970444099063, 8.26387826782729461898110954345, 9.060096967258951289025444272563, 10.34414032731487504981183897536, 11.375899367194721263214738705646, 11.8777162797403315151032371038, 12.85162349616596195922004002894, 13.963104777276605954087246213480, 14.91682953385721183356038139796, 15.68939745836947640313911227848, 17.2468708293268109352439748180, 17.932107925045023394471466060010, 19.04362956720762192819723263085, 19.91649702857930348990183615835, 20.35925868108554284184204083924, 21.648836842551272398058033667106, 22.11411414860347304849696835934, 23.42847051912443209281342848798, 23.93879015280543204732064568243, 24.77798231487749015217942058107

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