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Properties

Label 1-297-297.149-r0-0-0
Degree $1$
Conductor $297$
Sign $0.299 + 0.954i$
Analytic cond. $1.37926$
Root an. cond. $1.37926$
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  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.438 + 0.898i)5-s + (−0.848 + 0.529i)7-s + (−0.978 − 0.207i)8-s + (0.5 + 0.866i)10-s + (−0.961 + 0.275i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.997 + 0.0697i)20-s + (−0.766 + 0.642i)23-s + (−0.615 − 0.788i)25-s + (−0.309 + 0.951i)26-s + ⋯
L(s)  = 1  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.438 + 0.898i)5-s + (−0.848 + 0.529i)7-s + (−0.978 − 0.207i)8-s + (0.5 + 0.866i)10-s + (−0.961 + 0.275i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (−0.104 + 0.994i)17-s + (0.978 + 0.207i)19-s + (0.997 + 0.0697i)20-s + (−0.766 + 0.642i)23-s + (−0.615 − 0.788i)25-s + (−0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.299 + 0.954i$
Analytic conductor: \(1.37926\)
Root analytic conductor: \(1.37926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (0:\ ),\ 0.299 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5692834280 + 0.4181268754i\)
\(L(\frac12)\) \(\approx\) \(0.5692834280 + 0.4181268754i\)
\(L(1)\) \(\approx\) \(0.8906835033 - 0.08965537962i\)
\(L(1)\) \(\approx\) \(0.8906835033 - 0.08965537962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.559 - 0.829i)T \)
5 \( 1 + (-0.438 + 0.898i)T \)
7 \( 1 + (-0.848 + 0.529i)T \)
13 \( 1 + (-0.961 + 0.275i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.0348 + 0.999i)T \)
31 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.990 - 0.139i)T \)
61 \( 1 + (0.241 + 0.970i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (0.961 + 0.275i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.438 + 0.898i)T \)
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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

−24.82544718546999621953897626715, −24.50776795856867817898847117556, −23.4560446262244501991490563585, −22.660916923754643850127026738025, −21.99406630504595522279474643882, −20.59682755190564038290816463403, −20.09868680121156295562293683388, −18.86150540482382291744924603412, −17.57863751057514114346865602005, −16.75897499071763007906721568953, −16.061808703808333250187953751540, −15.355570717880275027909858203956, −14.058127895430167171383962924849, −13.31178388245367192557756630726, −12.397075557883455457046086737515, −11.6801422359680867935277627884, −9.88676566527873001464803608824, −9.079221289062341760927864823463, −7.80640494904683585404575413227, −7.177481951096962611996380726829, −5.895004174825637671766552642153, −4.85440453606066506586449875609, −3.981526825238623708736102384114, −2.77090587107347524254808421960, −0.36800196771569288515574994827, 1.90293878200425636452451527039, 3.06693418674971565657698962066, 3.76320842328676811014751371696, 5.212854898548795856613148462356, 6.2683729865927635663371420206, 7.286824923775148571202697302171, 8.860999814949010348957075049199, 9.94418382521797117381414037744, 10.61080118294629318174089341287, 11.89104901351671390990417263857, 12.311976119469220194424264215605, 13.55863377468017877923387255541, 14.48959053169937972742412481065, 15.26212953506507142786806374648, 16.17823650314665192328850527753, 17.777753521898247962226670501490, 18.60019633004544986131081956625, 19.52778221987380986958611968967, 19.847161050552624782094848680815, 21.38420043549534343878503396595, 22.07309939377539514722701095744, 22.59267543414119790417538203364, 23.58546201535129341308641168176, 24.41854529782441089580479372911, 25.69491169376346619340455508776

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