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Properties

Label 2-3-3.2-c42-0-8
Degree $2$
Conductor $3$
Sign $0.441 + 0.897i$
Analytic cond. $33.5183$
Root an. cond. $5.78950$
Motivic weight $42$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.69e6i·2-s + (−4.62e9 − 9.38e9i)3-s − 2.89e12·4-s − 1.56e14i·5-s + (2.53e16 − 1.24e16i)6-s + 1.13e17·7-s + 4.07e18i·8-s + (−6.66e19 + 8.67e19i)9-s + 4.22e20·10-s + 3.96e21i·11-s + (1.33e22 + 2.71e22i)12-s + 2.02e23·13-s + 3.06e23i·14-s + (−1.46e24 + 7.24e23i)15-s − 2.37e25·16-s − 8.78e25i·17-s + ⋯
L(s)  = 1  + 1.28i·2-s + (−0.441 − 0.897i)3-s − 0.657·4-s − 0.328i·5-s + (1.15 − 0.568i)6-s + 0.203·7-s + 0.441i·8-s + (−0.609 + 0.792i)9-s + 0.422·10-s + 0.536i·11-s + (0.290 + 0.589i)12-s + 0.821·13-s + 0.262i·14-s + (−0.294 + 0.145i)15-s − 1.22·16-s − 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(33.5183\)
Root analytic conductor: \(5.78950\)
Motivic weight: \(42\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :21),\ 0.441 + 0.897i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(0.9129633768\)
\(L(\frac12)\) \(\approx\) \(0.9129633768\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.62e9 + 9.38e9i)T \)
good2 \( 1 - 2.69e6iT - 4.39e12T^{2} \)
5 \( 1 + 1.56e14iT - 2.27e29T^{2} \)
7 \( 1 - 1.13e17T + 3.11e35T^{2} \)
11 \( 1 - 3.96e21iT - 5.47e43T^{2} \)
13 \( 1 - 2.02e23T + 6.10e46T^{2} \)
17 \( 1 + 8.78e25iT - 4.77e51T^{2} \)
19 \( 1 + 9.58e26T + 5.10e53T^{2} \)
23 \( 1 + 1.43e28iT - 1.55e57T^{2} \)
29 \( 1 + 5.25e30iT - 2.63e61T^{2} \)
31 \( 1 + 3.30e31T + 4.33e62T^{2} \)
37 \( 1 - 8.82e32T + 7.31e65T^{2} \)
41 \( 1 + 9.69e33iT - 5.45e67T^{2} \)
43 \( 1 - 8.56e32T + 4.03e68T^{2} \)
47 \( 1 + 1.68e34iT - 1.69e70T^{2} \)
53 \( 1 + 2.29e36iT - 2.62e72T^{2} \)
59 \( 1 + 2.54e37iT - 2.37e74T^{2} \)
61 \( 1 + 3.80e37T + 9.63e74T^{2} \)
67 \( 1 - 1.78e38T + 4.95e76T^{2} \)
71 \( 1 - 5.34e38iT - 5.66e77T^{2} \)
73 \( 1 + 2.20e39T + 1.81e78T^{2} \)
79 \( 1 + 3.32e39T + 5.01e79T^{2} \)
83 \( 1 + 2.39e40iT - 3.99e80T^{2} \)
89 \( 1 + 2.41e40iT - 7.48e81T^{2} \)
97 \( 1 - 3.93e41T + 2.78e83T^{2} \)
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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

−16.13502516334386326000124289713, −14.46944101713301437592972713251, −12.97688734001824579500637321461, −11.26290630874031606613533368435, −8.635173996077996840720860344508, −7.34598231572882538983709223983, −6.20648717236170220749472491085, −4.87518936242935063696927213267, −2.06611352719843624084303768182, −0.29745336374863631601716486733, 1.35552693331078962812557172638, 3.11632186307120682795261161663, 4.21301191592911341847244628709, 6.17594908669769478906867606468, 8.899728062895360034022540321325, 10.57205903989519819876731699345, 11.13510967963893444979496022044, 12.79300679091953125956490183877, 14.88738893878539309702961533528, 16.53468979952092827397464956086

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