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Properties

Label 1-291-291.110-r0-0-0
Degree $1$
Conductor $291$
Sign $0.933 - 0.358i$
Analytic cond. $1.35139$
Root an. cond. $1.35139$
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  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.0654 − 0.997i)5-s + (0.896 + 0.442i)7-s + (0.923 + 0.382i)8-s + (0.751 + 0.659i)10-s + (0.793 − 0.608i)11-s + (−0.997 − 0.0654i)13-s + (−0.896 + 0.442i)14-s + (−0.866 + 0.5i)16-s + (−0.442 − 0.896i)17-s + (0.831 + 0.555i)19-s + (−0.980 + 0.195i)20-s + i·22-s + (−0.946 − 0.321i)23-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.0654 − 0.997i)5-s + (0.896 + 0.442i)7-s + (0.923 + 0.382i)8-s + (0.751 + 0.659i)10-s + (0.793 − 0.608i)11-s + (−0.997 − 0.0654i)13-s + (−0.896 + 0.442i)14-s + (−0.866 + 0.5i)16-s + (−0.442 − 0.896i)17-s + (0.831 + 0.555i)19-s + (−0.980 + 0.195i)20-s + i·22-s + (−0.946 − 0.321i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ 0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9329143230 - 0.1727557871i\)
\(L(\frac12)\) \(\approx\) \(0.9329143230 - 0.1727557871i\)
\(L(1)\) \(\approx\) \(0.8578672123 + 0.03713553456i\)
\(L(1)\) \(\approx\) \(0.8578672123 + 0.03713553456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.608 + 0.793i)T \)
5 \( 1 + (0.0654 - 0.997i)T \)
7 \( 1 + (0.896 + 0.442i)T \)
11 \( 1 + (0.793 - 0.608i)T \)
13 \( 1 + (-0.997 - 0.0654i)T \)
17 \( 1 + (-0.442 - 0.896i)T \)
19 \( 1 + (0.831 + 0.555i)T \)
23 \( 1 + (-0.946 - 0.321i)T \)
29 \( 1 + (0.751 - 0.659i)T \)
31 \( 1 + (0.991 - 0.130i)T \)
37 \( 1 + (-0.321 - 0.946i)T \)
41 \( 1 + (-0.659 - 0.751i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.793 + 0.608i)T \)
59 \( 1 + (0.946 - 0.321i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.555 - 0.831i)T \)
71 \( 1 + (-0.659 + 0.751i)T \)
73 \( 1 + (-0.258 + 0.965i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (0.896 - 0.442i)T \)
89 \( 1 + (-0.923 - 0.382i)T \)
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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

−25.874824699762460341001894420917, −24.78070079019979183259460310248, −23.67726047540142273972783770167, −22.37820280177357545951383163867, −21.9870985409354563634266019329, −20.96844305055171265513437367100, −19.80302375962446064046096806590, −19.4693464472924195970693407416, −18.003056367450581487749839209, −17.720334880809108290578073087300, −16.80103491121787743240446914002, −15.305537623406439000455674469069, −14.35946223767544520546326049990, −13.520264742358943813040533529449, −12.04064692964588250950765166024, −11.53135181856562961860356952922, −10.39014567130732410448740212326, −9.83465693166794090851935747931, −8.500837345327982726422347494785, −7.47052420385910153260984671709, −6.70019640578041123184333529813, −4.79964382941675476187520130082, −3.78576309782763877580858851694, −2.51255928907499542165962850932, −1.47921730129764550108078447260, 0.85862541376403888905285828886, 2.14551787071859807793946506741, 4.35022220904188849635042386197, 5.184272031425198986653734133692, 6.10872938315706806890507352808, 7.500334360156133207547911960183, 8.33611574693165999953141391561, 9.14678378811610293645707647598, 10.00124414423585488253488974907, 11.47676275900407152506388101667, 12.218561295812307020704558947401, 13.83451599708853441725601924773, 14.31410190053427416728503872686, 15.58807639241893121563878122576, 16.28340377046305891955878016802, 17.28421053395659380498785144348, 17.82132940243135907707767436382, 18.98240225915028962401112968660, 19.88867376328600787049366345427, 20.72931331707365616033510795797, 21.90357482448011512237907787033, 22.89004115606398853284689693994, 24.2221112924856908289131673541, 24.57143792439760249880123288052, 25.02175143662835499047883383342

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