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Properties

Label 2-3-3.2-c20-0-5
Degree $2$
Conductor $3$
Sign $-0.439 - 0.898i$
Analytic cond. $7.60541$
Root an. cond. $2.75779$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.72e3i·2-s + (−2.59e4 − 5.30e4i)3-s − 1.92e6·4-s − 9.20e6i·5-s + (−9.14e7 + 4.47e7i)6-s + 4.08e8·7-s + 1.50e9i·8-s + (−2.14e9 + 2.75e9i)9-s − 1.58e10·10-s − 1.86e9i·11-s + (4.98e10 + 1.01e11i)12-s + 6.70e10·13-s − 7.04e11i·14-s + (−4.88e11 + 2.38e11i)15-s + 5.75e11·16-s − 1.72e12i·17-s + ⋯
L(s)  = 1  − 1.68i·2-s + (−0.439 − 0.898i)3-s − 1.83·4-s − 0.942i·5-s + (−1.51 + 0.739i)6-s + 1.44·7-s + 1.39i·8-s + (−0.613 + 0.789i)9-s − 1.58·10-s − 0.0718i·11-s + (0.804 + 1.64i)12-s + 0.486·13-s − 2.43i·14-s + (−0.846 + 0.414i)15-s + 0.523·16-s − 0.855i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.439 - 0.898i$
Analytic conductor: \(7.60541\)
Root analytic conductor: \(2.75779\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :10),\ -0.439 - 0.898i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.663038 + 1.06234i\)
\(L(\frac12)\) \(\approx\) \(0.663038 + 1.06234i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59e4 + 5.30e4i)T \)
good2 \( 1 + 1.72e3iT - 1.04e6T^{2} \)
5 \( 1 + 9.20e6iT - 9.53e13T^{2} \)
7 \( 1 - 4.08e8T + 7.97e16T^{2} \)
11 \( 1 + 1.86e9iT - 6.72e20T^{2} \)
13 \( 1 - 6.70e10T + 1.90e22T^{2} \)
17 \( 1 + 1.72e12iT - 4.06e24T^{2} \)
19 \( 1 + 8.30e12T + 3.75e25T^{2} \)
23 \( 1 - 1.84e13iT - 1.71e27T^{2} \)
29 \( 1 + 1.25e14iT - 1.76e29T^{2} \)
31 \( 1 - 7.73e14T + 6.71e29T^{2} \)
37 \( 1 - 3.58e14T + 2.31e31T^{2} \)
41 \( 1 + 1.14e16iT - 1.80e32T^{2} \)
43 \( 1 + 9.36e15T + 4.67e32T^{2} \)
47 \( 1 + 6.06e16iT - 2.76e33T^{2} \)
53 \( 1 - 1.40e17iT - 3.05e34T^{2} \)
59 \( 1 + 6.48e17iT - 2.61e35T^{2} \)
61 \( 1 + 1.26e18T + 5.08e35T^{2} \)
67 \( 1 - 1.97e18T + 3.32e36T^{2} \)
71 \( 1 - 1.85e18iT - 1.05e37T^{2} \)
73 \( 1 - 2.65e18T + 1.84e37T^{2} \)
79 \( 1 - 3.40e18T + 8.96e37T^{2} \)
83 \( 1 + 1.78e19iT - 2.40e38T^{2} \)
89 \( 1 - 3.26e19iT - 9.72e38T^{2} \)
97 \( 1 - 9.39e19T + 5.43e39T^{2} \)
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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

−20.24716463440565699171241740435, −18.66615752873997928436005461710, −17.34892860154249851870946799323, −13.63387321244717898840471659015, −12.21200163189869989175839362387, −11.05680764591416353953145494499, −8.495913724505337775136822820243, −4.80491283783909038642523704826, −1.93087955536826449541222952438, −0.74857935582946421197351590815, 4.55463135753605684958897262598, 6.29891509846053844376160253178, 8.363689626105504674052149911412, 10.84970008582520803166834030694, 14.46798644089759528493419244833, 15.21144515489752122892424442631, 16.97918489923515275179061491038, 18.09934947361486426789176285842, 21.31447343824337048363833923637, 22.84533459681917125523253576141

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