Properties

Label 2-3e2-9.7-c11-0-0
Degree $2$
Conductor $9$
Sign $-0.748 - 0.662i$
Analytic cond. $6.91508$
Root an. cond. $2.62965$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−19.4 + 33.7i)2-s + (5.51 − 420. i)3-s + (266. + 461. i)4-s + (1.69e3 + 2.92e3i)5-s + (1.40e4 + 8.37e3i)6-s + (−1.60e4 + 2.77e4i)7-s − 1.00e5·8-s + (−1.77e5 − 4.64e3i)9-s − 1.31e5·10-s + (−3.36e5 + 5.82e5i)11-s + (1.95e5 − 1.09e5i)12-s + (8.47e5 + 1.46e6i)13-s + (−6.23e5 − 1.07e6i)14-s + (1.24e6 − 6.95e5i)15-s + (1.40e6 − 2.44e6i)16-s − 5.41e6·17-s + ⋯
L(s)  = 1  + (−0.430 + 0.744i)2-s + (0.0130 − 0.999i)3-s + (0.130 + 0.225i)4-s + (0.242 + 0.419i)5-s + (0.739 + 0.439i)6-s + (−0.360 + 0.623i)7-s − 1.08·8-s + (−0.999 − 0.0261i)9-s − 0.416·10-s + (−0.629 + 1.08i)11-s + (0.227 − 0.127i)12-s + (0.633 + 1.09i)13-s + (−0.309 − 0.536i)14-s + (0.422 − 0.236i)15-s + (0.335 − 0.581i)16-s − 0.924·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 - 0.662i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-0.748 - 0.662i$
Analytic conductor: \(6.91508\)
Root analytic conductor: \(2.62965\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :11/2),\ -0.748 - 0.662i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.324395 + 0.856205i\)
\(L(\frac12)\) \(\approx\) \(0.324395 + 0.856205i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.51 + 420. i)T \)
good2 \( 1 + (19.4 - 33.7i)T + (-1.02e3 - 1.77e3i)T^{2} \)
5 \( 1 + (-1.69e3 - 2.92e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
7 \( 1 + (1.60e4 - 2.77e4i)T + (-9.88e8 - 1.71e9i)T^{2} \)
11 \( 1 + (3.36e5 - 5.82e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + (-8.47e5 - 1.46e6i)T + (-8.96e11 + 1.55e12i)T^{2} \)
17 \( 1 + 5.41e6T + 3.42e13T^{2} \)
19 \( 1 - 8.84e6T + 1.16e14T^{2} \)
23 \( 1 + (-2.61e6 - 4.52e6i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + (3.61e7 - 6.26e7i)T + (-6.10e15 - 1.05e16i)T^{2} \)
31 \( 1 + (1.03e8 + 1.78e8i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 - 7.22e8T + 1.77e17T^{2} \)
41 \( 1 + (2.14e8 + 3.71e8i)T + (-2.75e17 + 4.76e17i)T^{2} \)
43 \( 1 + (6.85e8 - 1.18e9i)T + (-4.64e17 - 8.04e17i)T^{2} \)
47 \( 1 + (-3.70e7 + 6.41e7i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 - 5.54e9T + 9.26e18T^{2} \)
59 \( 1 + (1.80e9 + 3.12e9i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (3.08e9 - 5.33e9i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (-3.11e9 - 5.39e9i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 + 1.73e10T + 2.31e20T^{2} \)
73 \( 1 + 1.89e10T + 3.13e20T^{2} \)
79 \( 1 + (-1.79e10 + 3.10e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 + (1.24e10 - 2.16e10i)T + (-6.43e20 - 1.11e21i)T^{2} \)
89 \( 1 - 7.94e10T + 2.77e21T^{2} \)
97 \( 1 + (3.32e10 - 5.76e10i)T + (-3.57e21 - 6.19e21i)T^{2} \)
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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

−18.50133228976089981900522237448, −17.94931047126853125531601143992, −16.37379077862304651660833304283, −14.90212535434002888389972567076, −13.09876920058505470552702272965, −11.67919573069372130794851242884, −9.066553424283897872264588118031, −7.40515141944737362958460523888, −6.23007113359897890169370786447, −2.41984515989539963781529930856, 0.55193526799254425515787763391, 3.19241541885879525181941030713, 5.62358782235415164174168156144, 8.801979911812366587467733929478, 10.24246697207475643518821857011, 11.18611709059725299176618918086, 13.40004390100601053575934915165, 15.31773675365417545308979532815, 16.51093155148087999140330590686, 18.18394077634682352301449733232

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