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Properties

Label 2-3e2-3.2-c8-0-1
Degree $2$
Conductor $9$
Sign $0.577 + 0.816i$
Analytic cond. $3.66640$
Root an. cond. $1.91478$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.24i·2-s + 238·4-s − 988. i·5-s + 1.65e3·7-s − 2.09e3i·8-s − 4.19e3·10-s + 2.13e4i·11-s − 2.62e4·13-s − 7.00e3i·14-s + 5.20e4·16-s − 1.51e4i·17-s + 4.66e4·19-s − 2.35e5i·20-s + 9.06e4·22-s + 3.28e5i·23-s + ⋯
L(s)  = 1  − 0.265i·2-s + 0.929·4-s − 1.58i·5-s + 0.688·7-s − 0.511i·8-s − 0.419·10-s + 1.45i·11-s − 0.919·13-s − 0.182i·14-s + 0.794·16-s − 0.181i·17-s + 0.357·19-s − 1.47i·20-s + 0.386·22-s + 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(3.66640\)
Root analytic conductor: \(1.91478\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :4),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.53969 - 0.797003i\)
\(L(\frac12)\) \(\approx\) \(1.53969 - 0.797003i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + 4.24iT - 256T^{2} \)
5 \( 1 + 988. iT - 3.90e5T^{2} \)
7 \( 1 - 1.65e3T + 5.76e6T^{2} \)
11 \( 1 - 2.13e4iT - 2.14e8T^{2} \)
13 \( 1 + 2.62e4T + 8.15e8T^{2} \)
17 \( 1 + 1.51e4iT - 6.97e9T^{2} \)
19 \( 1 - 4.66e4T + 1.69e10T^{2} \)
23 \( 1 - 3.28e5iT - 7.83e10T^{2} \)
29 \( 1 - 6.14e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.96e5T + 8.52e11T^{2} \)
37 \( 1 - 2.81e6T + 3.51e12T^{2} \)
41 \( 1 - 7.06e5iT - 7.98e12T^{2} \)
43 \( 1 + 2.21e6T + 1.16e13T^{2} \)
47 \( 1 + 1.63e6iT - 2.38e13T^{2} \)
53 \( 1 + 5.18e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.17e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.74e7T + 1.91e14T^{2} \)
67 \( 1 + 1.43e7T + 4.06e14T^{2} \)
71 \( 1 - 1.56e7iT - 6.45e14T^{2} \)
73 \( 1 + 8.90e6T + 8.06e14T^{2} \)
79 \( 1 - 3.27e7T + 1.51e15T^{2} \)
83 \( 1 + 8.50e7iT - 2.25e15T^{2} \)
89 \( 1 - 5.89e7iT - 3.93e15T^{2} \)
97 \( 1 + 2.44e7T + 7.83e15T^{2} \)
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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

−19.88128048563592376656692651939, −17.59442194777383088836091679751, −16.41174529581820502358707449725, −15.01144269951227098735604480782, −12.75865203634560429199435592488, −11.74151011440094799573884440791, −9.644242888902663933272328095892, −7.59476956713441111806571926180, −4.92784206874732720909095783752, −1.63018596306263491465773435880, 2.73750338537879341949836936406, 6.22974748587660468506079094773, 7.74053703127171466354489329985, 10.60025129642198135097972432621, 11.56685349254782266174603068421, 14.20501142008382155188432796360, 15.10883559580278349229824119286, 16.71617980069690576476837497665, 18.33929053281741577534652709849, 19.54184962159593855834933396899

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