Properties

Label 2-3e2-9.4-c3-0-1
Degree $2$
Conductor $9$
Sign $0.594 + 0.804i$
Analytic cond. $0.531017$
Root an. cond. $0.728709$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.18 − 3.78i)2-s + (3.55 + 3.78i)3-s + (−5.55 + 9.62i)4-s + (−2.31 + 4.00i)5-s + (6.55 − 21.7i)6-s + (−6.05 − 10.4i)7-s + 13.6·8-s + (−1.67 + 26.9i)9-s + 20.2·10-s + (−5.01 − 8.67i)11-s + (−56.2 + 13.2i)12-s + (24.2 − 42.0i)13-s + (−26.4 + 45.8i)14-s + (−23.4 + 5.49i)15-s + (14.6 + 25.4i)16-s + 75.3·17-s + ⋯
L(s)  = 1  + (−0.772 − 1.33i)2-s + (0.684 + 0.728i)3-s + (−0.694 + 1.20i)4-s + (−0.206 + 0.358i)5-s + (0.446 − 1.48i)6-s + (−0.327 − 0.566i)7-s + 0.602·8-s + (−0.0620 + 0.998i)9-s + 0.639·10-s + (−0.137 − 0.237i)11-s + (−1.35 + 0.317i)12-s + (0.518 − 0.897i)13-s + (−0.505 + 0.875i)14-s + (−0.402 + 0.0946i)15-s + (0.229 + 0.397i)16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.594 + 0.804i$
Analytic conductor: \(0.531017\)
Root analytic conductor: \(0.728709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :3/2),\ 0.594 + 0.804i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.600462 - 0.303034i\)
\(L(\frac12)\) \(\approx\) \(0.600462 - 0.303034i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.55 - 3.78i)T \)
good2 \( 1 + (2.18 + 3.78i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (2.31 - 4.00i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (6.05 + 10.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (5.01 + 8.67i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-24.2 + 42.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 75.3T + 4.91e3T^{2} \)
19 \( 1 + 116.T + 6.85e3T^{2} \)
23 \( 1 + (-19.0 + 32.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-11.3 - 19.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (15.0 - 26.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 130.T + 5.06e4T^{2} \)
41 \( 1 + (173. - 300. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (13.3 + 23.1i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (230. + 399. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 438.T + 1.48e5T^{2} \)
59 \( 1 + (4.18 - 7.24i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-41.0 - 71.0i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (341. - 591. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 - 470.T + 3.89e5T^{2} \)
79 \( 1 + (243. + 420. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (49.5 + 85.8i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 8.80T + 7.04e5T^{2} \)
97 \( 1 + (330. + 572. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−20.61914174957277648201216313855, −19.61476470405738677535335294555, −18.59801363309162403456432729925, −16.77693799993086464251421774646, −14.95846146462468485433937781861, −13.03963571238007264459722957608, −10.95181254561589137292804741136, −10.02099796895845103617255852559, −8.338707171049150906418840569156, −3.34962024508304345866916773268, 6.42275905042016689314401269882, 8.085664835665595463502938803439, 9.256610890634567397347525655757, 12.45166949290232207508515373311, 14.31845763182110684557309037513, 15.58090525747337205004237575777, 16.94023726446081188549837194070, 18.42978494776154011052373199594, 19.23697264098579876652278687290, 21.04125960132502301779273003424

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