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Properties

Label 2-3-3.2-c42-0-7
Degree $2$
Conductor $3$
Sign $-0.537 - 0.843i$
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  + 7.19e5i·2-s + (5.62e9 + 8.81e9i)3-s + 3.88e12·4-s + 4.30e14i·5-s + (−6.34e15 + 4.04e15i)6-s + 6.12e17·7-s + 5.95e18i·8-s + (−4.61e19 + 9.92e19i)9-s − 3.09e20·10-s + 4.42e21i·11-s + (2.18e22 + 3.42e22i)12-s + 2.30e23·13-s + 4.40e23i·14-s + (−3.79e24 + 2.42e24i)15-s + 1.27e25·16-s − 5.34e25i·17-s + ⋯
L(s)  = 1  + 0.342i·2-s + (0.537 + 0.843i)3-s + 0.882·4-s + 0.902i·5-s + (−0.289 + 0.184i)6-s + 1.09·7-s + 0.645i·8-s + (−0.421 + 0.906i)9-s − 0.309·10-s + 0.597i·11-s + (0.474 + 0.743i)12-s + 0.932·13-s + 0.376i·14-s + (−0.760 + 0.485i)15-s + 0.660·16-s − 0.773i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.537 - 0.843i$
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: \((2,\ 3,\ (\ :21),\ -0.537 - 0.843i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(3.590120901\)
\(L(\frac12)\) \(\approx\) \(3.590120901\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.62e9 - 8.81e9i)T \)
good2 \( 1 - 7.19e5iT - 4.39e12T^{2} \)
5 \( 1 - 4.30e14iT - 2.27e29T^{2} \)
7 \( 1 - 6.12e17T + 3.11e35T^{2} \)
11 \( 1 - 4.42e21iT - 5.47e43T^{2} \)
13 \( 1 - 2.30e23T + 6.10e46T^{2} \)
17 \( 1 + 5.34e25iT - 4.77e51T^{2} \)
19 \( 1 - 4.65e26T + 5.10e53T^{2} \)
23 \( 1 + 6.08e28iT - 1.55e57T^{2} \)
29 \( 1 - 2.44e30iT - 2.63e61T^{2} \)
31 \( 1 + 1.07e31T + 4.33e62T^{2} \)
37 \( 1 + 1.66e33T + 7.31e65T^{2} \)
41 \( 1 - 1.33e34iT - 5.45e67T^{2} \)
43 \( 1 + 3.17e33T + 4.03e68T^{2} \)
47 \( 1 + 2.20e35iT - 1.69e70T^{2} \)
53 \( 1 + 2.97e36iT - 2.62e72T^{2} \)
59 \( 1 - 4.22e36iT - 2.37e74T^{2} \)
61 \( 1 + 8.94e36T + 9.63e74T^{2} \)
67 \( 1 + 9.08e37T + 4.95e76T^{2} \)
71 \( 1 + 2.90e38iT - 5.66e77T^{2} \)
73 \( 1 - 1.55e39T + 1.81e78T^{2} \)
79 \( 1 + 6.10e39T + 5.01e79T^{2} \)
83 \( 1 - 1.82e40iT - 3.99e80T^{2} \)
89 \( 1 + 7.51e40iT - 7.48e81T^{2} \)
97 \( 1 - 3.76e41T + 2.78e83T^{2} \)
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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

−16.36486352926538935391319849953, −15.07068518239161808556478335374, −14.24800613752019246027312302023, −11.42098568508079038912188848373, −10.42511629515848330714381971037, −8.355691538020426761818268140748, −6.94040855045863719419496868120, −5.04276303990169939122715782369, −3.19496753961089897159115107418, −1.91081052428470281305480150632, 1.09575420191662641386239464217, 1.73378305462453579469954437658, 3.49725392401181606356192516968, 5.76304733777946235290677295297, 7.56007004342434590475737718981, 8.767705571707745153020973659719, 11.13584518117094060231837507714, 12.35054973076777965958327953984, 13.83153287932941960961591022833, 15.60693136978972853615448874499

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