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Properties

Label 1-1323-1323.59-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  + (−0.456 + 0.889i)2-s + (−0.583 − 0.811i)4-s + (−0.661 + 0.749i)5-s + (0.988 − 0.149i)8-s + (−0.365 − 0.930i)10-s + (0.998 − 0.0498i)11-s + (−0.921 + 0.388i)13-s + (−0.318 + 0.947i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (0.995 + 0.0995i)20-s + (−0.411 + 0.911i)22-s + (0.411 − 0.911i)23-s + (−0.124 − 0.992i)25-s + (0.0747 − 0.997i)26-s + ⋯
L(s)  = 1  + (−0.456 + 0.889i)2-s + (−0.583 − 0.811i)4-s + (−0.661 + 0.749i)5-s + (0.988 − 0.149i)8-s + (−0.365 − 0.930i)10-s + (0.998 − 0.0498i)11-s + (−0.921 + 0.388i)13-s + (−0.318 + 0.947i)16-s + (0.826 − 0.563i)17-s + (0.5 + 0.866i)19-s + (0.995 + 0.0995i)20-s + (−0.411 + 0.911i)22-s + (0.411 − 0.911i)23-s + (−0.124 − 0.992i)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: $\chi_{1323} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.863 + 0.504i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8995537579 + 0.2433256066i\)
\(L(\frac12)\) \(\approx\) \(0.8995537579 + 0.2433256066i\)
\(L(1)\) \(\approx\) \(0.7067628488 + 0.2849126409i\)
\(L(1)\) \(\approx\) \(0.7067628488 + 0.2849126409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.456 + 0.889i)T \)
5 \( 1 + (-0.661 + 0.749i)T \)
11 \( 1 + (0.998 - 0.0498i)T \)
13 \( 1 + (-0.921 + 0.388i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.411 - 0.911i)T \)
29 \( 1 + (0.583 - 0.811i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.661 + 0.749i)T \)
43 \( 1 + (0.980 - 0.198i)T \)
47 \( 1 + (-0.998 + 0.0498i)T \)
53 \( 1 + (-0.826 - 0.563i)T \)
59 \( 1 + (-0.853 + 0.521i)T \)
61 \( 1 + (0.583 - 0.811i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.826 - 0.563i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.921 + 0.388i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
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

−20.64667798472798730878047158075, −20.00737279791904801040863255401, −19.42035858350416165484520529397, −19.001624374983344948997307308352, −17.60029096788888846714416531858, −17.36676743346321664748391164906, −16.48670120964183200455906393585, −15.70978360010688519355231168378, −14.66642894290571922542486600844, −13.82307230105172924775345470749, −12.81501974103290526842461052125, −12.24149577376632169052938407431, −11.71578657748760780209195513316, −10.836773136118112421184958504060, −9.89424947799465542807524412414, −9.1570843185850680559727927823, −8.530495331479321264369520773981, −7.61126706950058035890247219663, −6.94013150566731920669759597070, −5.30656491724623194370897739037, −4.71355877086999253215257105429, −3.64966417058551255217342715839, −3.06295235248553671399055550642, −1.64023969403381627536442207588, −0.921576995572331189307501460002, 0.58158604542715566759545116813, 1.92970142404296306846181500779, 3.24213044300743080758611327615, 4.196574428315235373944449845182, 5.03669454229387402406888778818, 6.190985230358471779316043002490, 6.77652093167564867442139221582, 7.60881787482940779630293173997, 8.15321521965830399782564113627, 9.323815588379958000192682089086, 9.859332940202496412580322160127, 10.76362751670386178117970913028, 11.726189663156291652999110718758, 12.35193226072312399994937859088, 13.74059896430366017802829235117, 14.462432205459460603826079064344, 14.74577294566228279371617494918, 15.71927173862034316448287872678, 16.504343895387972803356547977955, 17.037660117711125962232372773198, 17.96643210825390695881580903532, 18.80053588133695118052928818299, 19.184931972901944532468680401024, 19.930359255781092953661643459597, 20.94174585788652222760225056512

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