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Properties

Label 2-1323-49.26-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.977 + 0.212i$
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  + (0.988 + 0.149i)4-s + (0.900 − 0.433i)7-s + (−0.590 − 1.22i)13-s + (0.955 + 0.294i)16-s + (−0.751 − 0.433i)19-s + (0.0747 + 0.997i)25-s + (0.955 − 0.294i)28-s + (−0.258 + 0.149i)31-s + (−1.95 + 0.294i)37-s + (0.367 + 1.61i)43-s + (0.623 − 0.781i)49-s + (−0.400 − 1.29i)52-s + (0.277 + 1.84i)61-s + (0.900 + 0.433i)64-s + (−0.955 − 1.65i)67-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)4-s + (0.900 − 0.433i)7-s + (−0.590 − 1.22i)13-s + (0.955 + 0.294i)16-s + (−0.751 − 0.433i)19-s + (0.0747 + 0.997i)25-s + (0.955 − 0.294i)28-s + (−0.258 + 0.149i)31-s + (−1.95 + 0.294i)37-s + (0.367 + 1.61i)43-s + (0.623 − 0.781i)49-s + (−0.400 − 1.29i)52-s + (0.277 + 1.84i)61-s + (0.900 + 0.433i)64-s + (−0.955 − 1.65i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.977 + 0.212i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.977 + 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421303482\)
\(L(\frac12)\) \(\approx\) \(1.421303482\)
\(L(1)\) 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 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (-0.988 - 0.149i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.590 + 1.22i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.751 + 0.433i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.258 - 0.149i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.95 - 0.294i)T + (0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.367 - 1.61i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (-0.277 - 1.84i)T + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.55 + 0.116i)T + (0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 + 0.930i)T^{2} \)
97 \( 1 + 1.12iT - T^{2} \)
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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

−10.04962234520465601710740320902, −8.853591502298725445654314482852, −7.978254446303637511663100647161, −7.43497619831103913161561883658, −6.66437685862603224072551722522, −5.59318785925493992600038706811, −4.83250637533588815489723918738, −3.58250117783194952180494501178, −2.59565390105041595052110660134, −1.45327481786846935970663607410, 1.78398681083938149565740415732, 2.35821044369960577354605247759, 3.80810020552439864540963910038, 4.85984344829945312047411447523, 5.73792212221145479269170013586, 6.66786156851768393133929816806, 7.29380671027178344175210555347, 8.253539403332729783474921507057, 8.932730529318404663160527133226, 9.996416286730401807275673372118

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