Properties

Label 2-1323-63.25-c1-0-9
Degree $2$
Conductor $1323$
Sign $0.799 - 0.600i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·2-s − 0.310·4-s + (1.76 − 3.05i)5-s + 3.00·8-s + (−2.29 + 3.96i)10-s + (0.589 + 1.02i)11-s + (1.61 + 2.78i)13-s − 3.28·16-s + (−2.45 + 4.24i)17-s + (3.43 + 5.94i)19-s + (−0.547 + 0.947i)20-s + (−0.765 − 1.32i)22-s + (−2.14 + 3.72i)23-s + (−3.71 − 6.43i)25-s + (−2.09 − 3.62i)26-s + ⋯
L(s)  = 1  − 0.919·2-s − 0.155·4-s + (0.788 − 1.36i)5-s + 1.06·8-s + (−0.724 + 1.25i)10-s + (0.177 + 0.307i)11-s + (0.446 + 0.773i)13-s − 0.820·16-s + (−0.594 + 1.02i)17-s + (0.787 + 1.36i)19-s + (−0.122 + 0.211i)20-s + (−0.163 − 0.282i)22-s + (−0.448 + 0.776i)23-s + (−0.743 − 1.28i)25-s + (−0.410 − 0.711i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.799 - 0.600i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.799 - 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9543821306\)
\(L(\frac12)\) \(\approx\) \(0.9543821306\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + 1.29T + 2T^{2} \)
5 \( 1 + (-1.76 + 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.589 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.61 - 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.45 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.43 - 5.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.14 - 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.36 - 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.92T + 31T^{2} \)
37 \( 1 + (-4.88 - 8.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.32 - 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.633T + 47T^{2} \)
53 \( 1 + (1.11 - 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.21T + 59T^{2} \)
61 \( 1 + 9.65T + 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + (0.519 - 0.898i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.00T + 79T^{2} \)
83 \( 1 + (-3.65 + 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.46 - 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.464893171868653161382757062091, −9.087280704395469129522899640531, −8.277870072362788855506602202693, −7.67259292950956400002813714220, −6.37706353611599148487326032393, −5.55715413951382239256467524594, −4.62046320739238916403341945748, −3.85105564801319199970875513148, −1.78858509636938415855785552757, −1.29087503743132345005885893187, 0.61274347642367248065322465500, 2.24279643567193322849432322508, 3.08762755507601780643695600867, 4.40564554398969219093322282940, 5.54233835943133909238470440454, 6.42527256872998245930679913616, 7.26062290710783626126756986686, 7.86545310341693279653564976807, 9.092442488563541899989538790745, 9.394371550297688972749046522809

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