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Properties

Label 2-1323-49.40-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.180 + 0.983i$
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  + (−0.0747 − 0.997i)4-s + (0.222 − 0.974i)7-s + (1.81 + 0.414i)13-s + (−0.988 + 0.149i)16-s + (−1.68 − 0.974i)19-s + (−0.733 + 0.680i)25-s + (−0.988 − 0.149i)28-s + (1.72 − 0.997i)31-s + (−0.0111 + 0.149i)37-s + (−1.19 − 1.49i)43-s + (−0.900 − 0.433i)49-s + (0.277 − 1.84i)52-s + (1.12 + 0.0841i)61-s + (0.222 + 0.974i)64-s + (0.988 + 1.71i)67-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)4-s + (0.222 − 0.974i)7-s + (1.81 + 0.414i)13-s + (−0.988 + 0.149i)16-s + (−1.68 − 0.974i)19-s + (−0.733 + 0.680i)25-s + (−0.988 − 0.149i)28-s + (1.72 − 0.997i)31-s + (−0.0111 + 0.149i)37-s + (−1.19 − 1.49i)43-s + (−0.900 − 0.433i)49-s + (0.277 − 1.84i)52-s + (1.12 + 0.0841i)61-s + (0.222 + 0.974i)64-s + (0.988 + 1.71i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075846865\)
\(L(\frac12)\) \(\approx\) \(1.075846865\)
\(L(1)\) 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 \( 1 + (0.0747 + 0.997i)T^{2} \)
5 \( 1 + (0.733 - 0.680i)T^{2} \)
11 \( 1 + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (-1.81 - 0.414i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (1.68 + 0.974i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.0111 - 0.149i)T + (-0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (-0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-1.12 - 0.0841i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.590 - 0.636i)T + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 - 0.589iT - T^{2} \)
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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

−9.828507970679896327810118952640, −8.761462467976784143889616795126, −8.257598920143090784653448289670, −6.88991208943601164591494960678, −6.46123214152476039422857128174, −5.52638943562908967838367873167, −4.42153385009983797655690119676, −3.83385641918393468580019432618, −2.15330192963795843913373422387, −0.991137069325056654341010841400, 1.85379443832162937236185699105, 3.01902831265171959151218343778, 3.88735371670369938120775058021, 4.82243758308665854434312827617, 6.14287271740877729442392936564, 6.45848012398651784117420106484, 8.035879818481224348354495169923, 8.291530304148654506061782976767, 8.862232654779607006235516407870, 9.998529549131960381130610885977

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