Properties

Label 2-1323-49.47-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.942 - 0.335i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
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  + (−0.826 − 0.563i)4-s + (0.222 + 0.974i)7-s + (−0.290 + 0.0663i)13-s + (0.365 + 0.930i)16-s + (1.68 + 0.974i)19-s + (0.955 − 0.294i)25-s + (0.365 − 0.930i)28-s + (0.975 − 0.563i)31-s + (−1.36 + 0.930i)37-s + (0.914 − 1.14i)43-s + (−0.900 + 0.433i)49-s + (0.277 + 0.108i)52-s + (1.12 + 1.64i)61-s + (0.222 − 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)4-s + (0.222 + 0.974i)7-s + (−0.290 + 0.0663i)13-s + (0.365 + 0.930i)16-s + (1.68 + 0.974i)19-s + (0.955 − 0.294i)25-s + (0.365 − 0.930i)28-s + (0.975 − 0.563i)31-s + (−1.36 + 0.930i)37-s + (0.914 − 1.14i)43-s + (−0.900 + 0.433i)49-s + (0.277 + 0.108i)52-s + (1.12 + 1.64i)61-s + (0.222 − 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.942 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9237606852\)
\(L(\frac12)\) \(\approx\) \(0.9237606852\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.222 - 0.974i)T \)
good2 \( 1 + (0.826 + 0.563i)T^{2} \)
5 \( 1 + (-0.955 + 0.294i)T^{2} \)
11 \( 1 + (0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.290 - 0.0663i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.988 - 0.149i)T^{2} \)
19 \( 1 + (-1.68 - 0.974i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.975 + 0.563i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.36 - 0.930i)T + (0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (-1.12 - 1.64i)T + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.255 - 0.829i)T + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 + 1.36iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.881747358662268316654978905816, −9.023679238061335597644171027256, −8.471362025940182723911915456803, −7.55920241052307615321388795707, −6.41060011562060407138241285792, −5.48442292384963688751074859250, −5.04545239566626612603419144522, −3.90780681207675953569673776379, −2.72279859696078191042970372981, −1.33064800379332359694997552608, 0.985723670048946161206812940610, 2.88361662340205635903268130157, 3.71636542627490656687121619357, 4.74615908787407314768066254422, 5.25872408813903324047767747365, 6.76523993304005520299763098574, 7.41419678677991474660651531961, 8.105586055836899190781623387506, 9.051372175287564405865684301322, 9.647045546019426039142752429330

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