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Properties

Label 2-3-3.2-c22-0-1
Degree $2$
Conductor $3$
Sign $-0.956 + 0.292i$
Analytic cond. $9.20122$
Root an. cond. $3.03335$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.38e3i·2-s + (1.69e5 − 5.17e4i)3-s − 7.23e6·4-s + 6.36e7i·5-s + (1.75e8 + 5.72e8i)6-s − 2.60e9·7-s − 1.02e10i·8-s + (2.60e10 − 1.75e10i)9-s − 2.15e11·10-s + 2.74e10i·11-s + (−1.22e12 + 3.74e11i)12-s − 6.09e9·13-s − 8.80e12i·14-s + (3.29e12 + 1.07e13i)15-s + 4.44e12·16-s + 8.12e12i·17-s + ⋯
L(s)  = 1  + 1.65i·2-s + (0.956 − 0.292i)3-s − 1.72·4-s + 1.30i·5-s + (0.482 + 1.57i)6-s − 1.31·7-s − 1.19i·8-s + (0.829 − 0.559i)9-s − 2.15·10-s + 0.0963i·11-s + (−1.65 + 0.504i)12-s − 0.00340·13-s − 2.17i·14-s + (0.381 + 1.24i)15-s + 0.252·16-s + 0.237i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.956 + 0.292i$
Analytic conductor: \(9.20122\)
Root analytic conductor: \(3.03335\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :11),\ -0.956 + 0.292i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(0.242556 - 1.62312i\)
\(L(\frac12)\) \(\approx\) \(0.242556 - 1.62312i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69e5 + 5.17e4i)T \)
good2 \( 1 - 3.38e3iT - 4.19e6T^{2} \)
5 \( 1 - 6.36e7iT - 2.38e15T^{2} \)
7 \( 1 + 2.60e9T + 3.90e18T^{2} \)
11 \( 1 - 2.74e10iT - 8.14e22T^{2} \)
13 \( 1 + 6.09e9T + 3.21e24T^{2} \)
17 \( 1 - 8.12e12iT - 1.17e27T^{2} \)
19 \( 1 - 7.54e13T + 1.35e28T^{2} \)
23 \( 1 - 1.66e15iT - 9.07e29T^{2} \)
29 \( 1 - 1.26e16iT - 1.48e32T^{2} \)
31 \( 1 - 1.69e16T + 6.45e32T^{2} \)
37 \( 1 - 1.59e17T + 3.16e34T^{2} \)
41 \( 1 - 4.83e17iT - 3.02e35T^{2} \)
43 \( 1 - 4.91e17T + 8.63e35T^{2} \)
47 \( 1 - 7.73e17iT - 6.11e36T^{2} \)
53 \( 1 + 5.90e18iT - 8.59e37T^{2} \)
59 \( 1 + 2.68e19iT - 9.09e38T^{2} \)
61 \( 1 - 2.43e19T + 1.89e39T^{2} \)
67 \( 1 + 1.30e20T + 1.49e40T^{2} \)
71 \( 1 + 4.07e20iT - 5.34e40T^{2} \)
73 \( 1 + 4.04e20T + 9.84e40T^{2} \)
79 \( 1 - 1.11e21T + 5.59e41T^{2} \)
83 \( 1 - 1.31e19iT - 1.65e42T^{2} \)
89 \( 1 + 2.62e20iT - 7.70e42T^{2} \)
97 \( 1 + 1.92e21T + 5.11e43T^{2} \)
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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

−22.22694520577990157741700890449, −19.39345780543958100440069826780, −18.06130619100091397186703377482, −15.90242903742294102579776918005, −14.74565928857942166186961036424, −13.44733135813779274291068303824, −9.539657149998931929046855752156, −7.50144732091555667112179073425, −6.41172389356607934760918331858, −3.24423073600163050121881869767, 0.74115108378253705518423164001, 2.68081540151200740157366115719, 4.26724804491130504538189631432, 8.869254268699354946280466420355, 9.976103913997127153057445156112, 12.43736317189804011858670026576, 13.45942763573570414593560815061, 16.19988945059009628040831446757, 18.95077428531659782550424973125, 20.06085581866702325648046053752

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