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Properties

Label 1-1323-1323.1181-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.863 - 0.504i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (−0.542 + 0.840i)2-s + (−0.411 − 0.911i)4-s + (0.980 − 0.198i)5-s + (0.988 + 0.149i)8-s + (−0.365 + 0.930i)10-s + (−0.456 − 0.889i)11-s + (0.797 − 0.603i)13-s + (−0.661 + 0.749i)16-s + (0.826 + 0.563i)17-s + (0.5 − 0.866i)19-s + (−0.583 − 0.811i)20-s + (0.995 + 0.0995i)22-s + (−0.995 − 0.0995i)23-s + (0.921 − 0.388i)25-s + (0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (−0.542 + 0.840i)2-s + (−0.411 − 0.911i)4-s + (0.980 − 0.198i)5-s + (0.988 + 0.149i)8-s + (−0.365 + 0.930i)10-s + (−0.456 − 0.889i)11-s + (0.797 − 0.603i)13-s + (−0.661 + 0.749i)16-s + (0.826 + 0.563i)17-s + (0.5 − 0.866i)19-s + (−0.583 − 0.811i)20-s + (0.995 + 0.0995i)22-s + (−0.995 − 0.0995i)23-s + (0.921 − 0.388i)25-s + (0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.863 - 0.504i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.863 - 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249042405 - 0.3378608541i\)
\(L(\frac12)\) \(\approx\) \(1.249042405 - 0.3378608541i\)
\(L(1)\) \(\approx\) \(0.9556949667 + 0.08233731475i\)
\(L(1)\) \(\approx\) \(0.9556949667 + 0.08233731475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.542 + 0.840i)T \)
5 \( 1 + (0.980 - 0.198i)T \)
11 \( 1 + (-0.456 - 0.889i)T \)
13 \( 1 + (0.797 - 0.603i)T \)
17 \( 1 + (0.826 + 0.563i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.995 - 0.0995i)T \)
29 \( 1 + (0.411 - 0.911i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.980 - 0.198i)T \)
43 \( 1 + (-0.318 - 0.947i)T \)
47 \( 1 + (0.456 + 0.889i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.878 - 0.478i)T \)
61 \( 1 + (0.411 - 0.911i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.826 + 0.563i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.797 - 0.603i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−20.98585298205237877410325296083, −20.38153499066324303276913009743, −19.51036854948005690710543281233, −18.46369510273991435243639152163, −18.153467619480435090789285435229, −17.56055394700151266036659824022, −16.405434801960755544869986051084, −16.12459025080525176715624317998, −14.54767284778060243817627634071, −14.019913859378966840570344741544, −13.18944702152564466400539297898, −12.4076488877941356856662898648, −11.73366345479366243169903471030, −10.6557024993403741280844494495, −10.13509554129213854098189123895, −9.45732967002093119742464022917, −8.70120712940395246321513488599, −7.67107572176198178408470728436, −6.92962374320658494204832101131, −5.7714975237900058779082563086, −4.90221364032000244589563540677, −3.784677926194767158212132524845, −2.925873679593261105359984777507, −1.896477278827129694295718780695, −1.32507345374653723222118939455, 0.645752288120596906998775085786, 1.6203564988512070730197291221, 2.806768605074171357360457433193, 4.05907182399950141879862845888, 5.31914757296202912498337517980, 5.771766308479494494837074498, 6.41231572935056834682682620263, 7.57739372049198487350719708335, 8.314223188669392112090174294436, 8.99126334097801883075062258134, 9.897585991215761385684803797358, 10.48287874681962740508784915715, 11.2982816466980534538733007520, 12.65930649507636832089812818730, 13.50437852791598700502867168078, 13.924857310915786231454871545171, 14.79009777089573089042142821162, 15.87688582037464004573150196095, 16.14256430249186910053104457827, 17.341263989552446926827871824231, 17.53156926424895568795157939229, 18.59801876982834543545157857680, 18.94670172194929628960716225981, 20.14134323777732506358475171783, 20.77257314902142388550252703790

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