Properties

Label 2-3e2-9.2-c6-0-1
Degree $2$
Conductor $9$
Sign $0.626 - 0.779i$
Analytic cond. $2.07048$
Root an. cond. $1.43891$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.96 + 4.02i)2-s + (0.282 + 26.9i)3-s + (0.352 + 0.610i)4-s + (80.3 − 46.3i)5-s + (−106. + 189. i)6-s + (60.0 − 103. i)7-s − 509. i·8-s + (−728. + 15.2i)9-s + 746.·10-s + (−1.29e3 − 750. i)11-s + (−16.3 + 9.69i)12-s + (2.14e3 + 3.71e3i)13-s + (836. − 482. i)14-s + (1.27e3 + 2.15e3i)15-s + (2.07e3 − 3.58e3i)16-s + 940. i·17-s + ⋯
L(s)  = 1  + (0.870 + 0.502i)2-s + (0.0104 + 0.999i)3-s + (0.00550 + 0.00954i)4-s + (0.642 − 0.370i)5-s + (−0.493 + 0.875i)6-s + (0.174 − 0.303i)7-s − 0.994i·8-s + (−0.999 + 0.0209i)9-s + 0.746·10-s + (−0.976 − 0.563i)11-s + (−0.00948 + 0.00560i)12-s + (0.975 + 1.68i)13-s + (0.304 − 0.175i)14-s + (0.377 + 0.638i)15-s + (0.505 − 0.875i)16-s + 0.191i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(2.07048\)
Root analytic conductor: \(1.43891\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :3),\ 0.626 - 0.779i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.65660 + 0.793686i\)
\(L(\frac12)\) \(\approx\) \(1.65660 + 0.793686i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.282 - 26.9i)T \)
good2 \( 1 + (-6.96 - 4.02i)T + (32 + 55.4i)T^{2} \)
5 \( 1 + (-80.3 + 46.3i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-60.0 + 103. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (1.29e3 + 750. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-2.14e3 - 3.71e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 940. iT - 2.41e7T^{2} \)
19 \( 1 + 8.39e3T + 4.70e7T^{2} \)
23 \( 1 + (-7.82e3 + 4.51e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (1.16e4 + 6.70e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-6.75e3 - 1.16e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 3.90e4T + 2.56e9T^{2} \)
41 \( 1 + (-9.52e4 + 5.50e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (5.77e3 - 9.99e3i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-2.28e4 - 1.31e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 - 1.29e4iT - 2.21e10T^{2} \)
59 \( 1 + (1.60e5 - 9.29e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-4.87e4 + 8.43e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-4.83e4 - 8.37e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 2.64e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.54e5T + 1.51e11T^{2} \)
79 \( 1 + (-4.31e4 + 7.48e4i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (5.43e5 + 3.14e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + 8.74e5iT - 4.96e11T^{2} \)
97 \( 1 + (-9.51e4 + 1.64e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−21.07532639863319730848673641720, −18.92354507481479261412668675964, −16.87356115693668899347088262276, −15.79222939993606328167359670452, −14.35421661628409684420615950355, −13.28980541110725874982955268944, −10.80408479504958945658562640110, −9.100857943142000633685803207732, −5.96022311335153170597818618642, −4.34009224583379823153072891146, 2.58024510002016779067401278153, 5.69778240647139513358871047649, 8.104518337824116500198271611699, 10.91653166298101462533526442220, 12.70825536671756957849324616603, 13.37915068344276813344295768320, 14.93613394448637830102419810471, 17.52927270778987105710606912668, 18.32102273990810621392211445293, 20.19953573053824299588509608003

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