Properties

Label 2-3e2-9.7-c11-0-7
Degree $2$
Conductor $9$
Sign $0.898 + 0.438i$
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  + (−36.0 + 62.3i)2-s + (417. − 51.3i)3-s + (−1.57e3 − 2.72e3i)4-s + (−5.31e3 − 9.19e3i)5-s + (−1.18e4 + 2.79e4i)6-s + (2.65e4 − 4.60e4i)7-s + 7.88e4·8-s + (1.71e5 − 4.28e4i)9-s + 7.65e5·10-s + (−2.41e5 + 4.17e5i)11-s + (−7.96e5 − 1.05e6i)12-s + (−7.41e5 − 1.28e6i)13-s + (1.91e6 + 3.31e6i)14-s + (−2.69e6 − 3.57e6i)15-s + (3.76e5 − 6.52e5i)16-s + 7.33e5·17-s + ⋯
L(s)  = 1  + (−0.796 + 1.37i)2-s + (0.992 − 0.121i)3-s + (−0.767 − 1.32i)4-s + (−0.760 − 1.31i)5-s + (−0.622 + 1.46i)6-s + (0.597 − 1.03i)7-s + 0.851·8-s + (0.970 − 0.241i)9-s + 2.42·10-s + (−0.451 + 0.781i)11-s + (−0.923 − 1.22i)12-s + (−0.553 − 0.958i)13-s + (0.951 + 1.64i)14-s + (−0.914 − 1.21i)15-s + (0.0897 − 0.155i)16-s + 0.125·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.898 + 0.438i$
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: \((2,\ 9,\ (\ :11/2),\ 0.898 + 0.438i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.08275 - 0.249940i\)
\(L(\frac12)\) \(\approx\) \(1.08275 - 0.249940i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-417. + 51.3i)T \)
good2 \( 1 + (36.0 - 62.3i)T + (-1.02e3 - 1.77e3i)T^{2} \)
5 \( 1 + (5.31e3 + 9.19e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
7 \( 1 + (-2.65e4 + 4.60e4i)T + (-9.88e8 - 1.71e9i)T^{2} \)
11 \( 1 + (2.41e5 - 4.17e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + (7.41e5 + 1.28e6i)T + (-8.96e11 + 1.55e12i)T^{2} \)
17 \( 1 - 7.33e5T + 3.42e13T^{2} \)
19 \( 1 - 6.31e6T + 1.16e14T^{2} \)
23 \( 1 + (1.75e7 + 3.03e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + (3.77e6 - 6.53e6i)T + (-6.10e15 - 1.05e16i)T^{2} \)
31 \( 1 + (8.05e7 + 1.39e8i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 - 3.37e8T + 1.77e17T^{2} \)
41 \( 1 + (-5.58e8 - 9.67e8i)T + (-2.75e17 + 4.76e17i)T^{2} \)
43 \( 1 + (2.46e8 - 4.27e8i)T + (-4.64e17 - 8.04e17i)T^{2} \)
47 \( 1 + (-4.21e8 + 7.29e8i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + 2.78e9T + 9.26e18T^{2} \)
59 \( 1 + (3.60e9 + 6.23e9i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (1.73e9 - 3.01e9i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (-8.70e9 - 1.50e10i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 - 8.70e9T + 2.31e20T^{2} \)
73 \( 1 - 2.17e10T + 3.13e20T^{2} \)
79 \( 1 + (1.02e10 - 1.77e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 + (-9.52e9 + 1.64e10i)T + (-6.43e20 - 1.11e21i)T^{2} \)
89 \( 1 - 3.48e10T + 2.77e21T^{2} \)
97 \( 1 + (-3.72e10 + 6.44e10i)T + (-3.57e21 - 6.19e21i)T^{2} \)
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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

−18.16141610837771992130038578919, −16.84885221078110209927477600712, −15.69889699597745836325476263937, −14.53840890454380529927220195394, −12.77545857083733595400845678014, −9.734254425043148763756029845490, −8.124685248318696297231810282284, −7.57774411097221134677026712027, −4.64500331682267686641824769631, −0.72850785257533687107215754472, 2.19357707715461120577912974215, 3.39917365211498809390532078321, 7.80567273627285577911336964761, 9.250934000299204024509095847102, 10.85535655184901299311817841765, 11.99324091489431199927986325773, 14.20793110636654426159908344114, 15.53876124097118223829986844197, 18.25886726977406429690092428291, 18.87571851668767147862645210496

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