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Properties

Label 1-1323-1323.572-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 + (−0.318 − 0.947i)5-s + (0.988 − 0.149i)8-s + (−0.623 + 0.781i)10-s + (−0.542 − 0.840i)11-s + (−0.921 + 0.388i)13-s + (−0.661 − 0.749i)16-s + (−0.900 − 0.433i)17-s − 19-s + (0.995 + 0.0995i)20-s + (−0.411 + 0.911i)22-s + (0.583 + 0.811i)23-s + (−0.797 + 0.603i)25-s + (0.826 + 0.563i)26-s + ⋯
L(s)  = 1  + (−0.542 − 0.840i)2-s + (−0.411 + 0.911i)4-s + (−0.318 − 0.947i)5-s + (0.988 − 0.149i)8-s + (−0.623 + 0.781i)10-s + (−0.542 − 0.840i)11-s + (−0.921 + 0.388i)13-s + (−0.661 − 0.749i)16-s + (−0.900 − 0.433i)17-s − 19-s + (0.995 + 0.0995i)20-s + (−0.411 + 0.911i)22-s + (0.583 + 0.811i)23-s + (−0.797 + 0.603i)25-s + (0.826 + 0.563i)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: $\chi_{1323} (572, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.863 + 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3150285564 + 0.08521393406i\)
\(L(\frac12)\) \(\approx\) \(0.3150285564 + 0.08521393406i\)
\(L(1)\) \(\approx\) \(0.5092044412 - 0.2561771123i\)
\(L(1)\) \(\approx\) \(0.5092044412 - 0.2561771123i\)

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 \( 1 + (-0.318 - 0.947i)T \)
11 \( 1 + (-0.542 - 0.840i)T \)
13 \( 1 + (-0.921 + 0.388i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.583 + 0.811i)T \)
29 \( 1 + (0.583 - 0.811i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (-0.661 + 0.749i)T \)
43 \( 1 + (0.980 - 0.198i)T \)
47 \( 1 + (0.456 - 0.889i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
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 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.921 + 0.388i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−20.66815595240345187945670587006, −19.876349294782052551079225763201, −19.15297191167005928310228541941, −18.55980057903838066742266394356, −17.6472262322179206551676058203, −17.33842792557474484458311583764, −16.21833200479200995784663428045, −15.45661875353223459296657704056, −14.76278989735590289470868570292, −14.52605450373849796325080334034, −13.21178052559879158724840834996, −12.56928464482198671529304059459, −11.27007310643911741905545946637, −10.52169520777589995485665096726, −10.05509081638784583181410392215, −9.00301596594340930521710280959, −8.148647103401548376008365361382, −7.331945085165982833632284211345, −6.81009515172020008247846129497, −5.97578195629338352779907698242, −4.85449793622481587375566641879, −4.19417550381369003928786283101, −2.720484475435119995381593111080, −1.94885285676460591895485937530, −0.18897669666410821393509830134, 0.89175083554182233397914339137, 2.06621195466750394902091551531, 2.89122960859144553349845340840, 4.07096139232751414359320780457, 4.673103629113124336142881551845, 5.65167212841491104795606607275, 7.05456154937844587053333486105, 7.81621079131226861009329132118, 8.73123331246409745696146663953, 9.11566754286720054757426109708, 10.09977293555740707576837066914, 10.99681702987012264816067064964, 11.65870890929613756912651511365, 12.39972718082117891431455787593, 13.19810088937665522539151692896, 13.67575515008869438369381406317, 14.96393785927727792770830232293, 15.95625188230468180456573497211, 16.578015140882201794055364624404, 17.26485609279667720861384707887, 17.97259075487924887745617790468, 19.00208013466040086626504799250, 19.508648930159295520942861646938, 20.09369371868010794821889269643, 21.03556236128947959259855271420

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