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Properties

Label 2-3-3.2-c42-0-1
Degree $2$
Conductor $3$
Sign $-0.879 + 0.476i$
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  + 1.88e6i·2-s + (9.19e9 − 4.98e9i)3-s + 8.55e11·4-s + 4.92e14i·5-s + (9.38e15 + 1.73e16i)6-s − 8.30e17·7-s + 9.88e18i·8-s + (5.97e19 − 9.16e19i)9-s − 9.26e20·10-s − 6.44e21i·11-s + (7.86e21 − 4.26e21i)12-s − 2.92e23·13-s − 1.56e24i·14-s + (2.45e24 + 4.52e24i)15-s − 1.48e25·16-s + 1.05e26i·17-s + ⋯
L(s)  = 1  + 0.897i·2-s + (0.879 − 0.476i)3-s + 0.194·4-s + 1.03i·5-s + (0.427 + 0.789i)6-s − 1.48·7-s + 1.07i·8-s + (0.545 − 0.837i)9-s − 0.926·10-s − 0.871i·11-s + (0.170 − 0.0927i)12-s − 1.18·13-s − 1.33i·14-s + (0.492 + 0.907i)15-s − 0.767·16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.476i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.879 + 0.476i$
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: \((2,\ 3,\ (\ :21),\ -0.879 + 0.476i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(0.9207633188\)
\(L(\frac12)\) \(\approx\) \(0.9207633188\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-9.19e9 + 4.98e9i)T \)
good2 \( 1 - 1.88e6iT - 4.39e12T^{2} \)
5 \( 1 - 4.92e14iT - 2.27e29T^{2} \)
7 \( 1 + 8.30e17T + 3.11e35T^{2} \)
11 \( 1 + 6.44e21iT - 5.47e43T^{2} \)
13 \( 1 + 2.92e23T + 6.10e46T^{2} \)
17 \( 1 - 1.05e26iT - 4.77e51T^{2} \)
19 \( 1 + 7.89e26T + 5.10e53T^{2} \)
23 \( 1 + 3.44e28iT - 1.55e57T^{2} \)
29 \( 1 - 3.85e30iT - 2.63e61T^{2} \)
31 \( 1 + 7.97e29T + 4.33e62T^{2} \)
37 \( 1 + 7.71e32T + 7.31e65T^{2} \)
41 \( 1 - 1.17e33iT - 5.45e67T^{2} \)
43 \( 1 + 2.26e34T + 4.03e68T^{2} \)
47 \( 1 + 9.68e34iT - 1.69e70T^{2} \)
53 \( 1 - 1.63e36iT - 2.62e72T^{2} \)
59 \( 1 + 1.67e37iT - 2.37e74T^{2} \)
61 \( 1 + 1.58e37T + 9.63e74T^{2} \)
67 \( 1 - 1.30e38T + 4.95e76T^{2} \)
71 \( 1 - 1.38e39iT - 5.66e77T^{2} \)
73 \( 1 + 2.56e38T + 1.81e78T^{2} \)
79 \( 1 - 1.85e39T + 5.01e79T^{2} \)
83 \( 1 - 1.31e40iT - 3.99e80T^{2} \)
89 \( 1 - 3.99e40iT - 7.48e81T^{2} \)
97 \( 1 + 2.91e41T + 2.78e83T^{2} \)
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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

−16.86205165355227995076817570914, −15.28519031307722557690185767781, −14.36055024590603690817569465919, −12.68167038443476368252718596122, −10.39571085973526172869262547422, −8.498069745160832668357796992254, −6.93970519675733162794458163827, −6.30141611803134740632732133979, −3.35581564355856892867782863304, −2.31453160007642818196212457474, 0.21482245399027455688721877390, 2.02984955865193224190853266809, 3.15594008515565554312107982193, 4.62548270007352095973690826232, 7.13693707288084326956985415912, 9.312067133668760786962020886012, 9.993498990787983104956710360821, 12.23645506902314923648407518108, 13.22288984335437358311015311642, 15.43304376312394599380421028596

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