Properties

Label 2-3-3.2-c32-0-4
Degree $2$
Conductor $3$
Sign $-0.540 + 0.841i$
Analytic cond. $19.4599$
Root an. cond. $4.41134$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.03e5i·2-s + (−2.32e7 + 3.62e7i)3-s − 6.48e9·4-s + 2.49e11i·5-s + (3.76e12 + 2.41e12i)6-s − 3.04e13·7-s + 2.27e14i·8-s + (−7.71e14 − 1.68e15i)9-s + 2.59e16·10-s − 4.78e16i·11-s + (1.50e17 − 2.34e17i)12-s + 5.67e17·13-s + 3.16e18i·14-s + (−9.05e18 − 5.80e18i)15-s − 4.25e18·16-s − 9.91e18i·17-s + ⋯
L(s)  = 1  − 1.58i·2-s + (−0.540 + 0.841i)3-s − 1.50·4-s + 1.63i·5-s + (1.33 + 0.855i)6-s − 0.915·7-s + 0.807i·8-s + (−0.416 − 0.909i)9-s + 2.59·10-s − 1.04i·11-s + (0.815 − 1.27i)12-s + 0.852·13-s + 1.45i·14-s + (−1.37 − 0.884i)15-s − 0.230·16-s − 0.203i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.540 + 0.841i$
Analytic conductor: \(19.4599\)
Root analytic conductor: \(4.41134\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :16),\ -0.540 + 0.841i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.448668 - 0.821186i\)
\(L(\frac12)\) \(\approx\) \(0.448668 - 0.821186i\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.32e7 - 3.62e7i)T \)
good2 \( 1 + 1.03e5iT - 4.29e9T^{2} \)
5 \( 1 - 2.49e11iT - 2.32e22T^{2} \)
7 \( 1 + 3.04e13T + 1.10e27T^{2} \)
11 \( 1 + 4.78e16iT - 2.11e33T^{2} \)
13 \( 1 - 5.67e17T + 4.42e35T^{2} \)
17 \( 1 + 9.91e18iT - 2.36e39T^{2} \)
19 \( 1 - 3.55e20T + 8.31e40T^{2} \)
23 \( 1 + 3.07e20iT - 3.76e43T^{2} \)
29 \( 1 + 3.31e23iT - 6.26e46T^{2} \)
31 \( 1 - 3.59e23T + 5.29e47T^{2} \)
37 \( 1 - 3.31e24T + 1.52e50T^{2} \)
41 \( 1 + 3.35e25iT - 4.06e51T^{2} \)
43 \( 1 - 1.34e26T + 1.86e52T^{2} \)
47 \( 1 - 9.71e23iT - 3.21e53T^{2} \)
53 \( 1 + 5.82e27iT - 1.50e55T^{2} \)
59 \( 1 + 1.32e28iT - 4.64e56T^{2} \)
61 \( 1 + 6.55e28T + 1.35e57T^{2} \)
67 \( 1 + 8.95e28T + 2.71e58T^{2} \)
71 \( 1 + 4.59e28iT - 1.73e59T^{2} \)
73 \( 1 - 2.56e28T + 4.22e59T^{2} \)
79 \( 1 - 3.75e30T + 5.29e60T^{2} \)
83 \( 1 - 1.44e30iT - 2.57e61T^{2} \)
89 \( 1 + 4.42e30iT - 2.40e62T^{2} \)
97 \( 1 - 7.67e31T + 3.77e63T^{2} \)
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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

−18.19296635157294595701157399777, −15.83358632677572405611982473308, −13.78310378759570794155837286221, −11.60405456400427261620048001842, −10.70965759581178684836872165024, −9.605958763119258228101545075209, −6.22734236307828867942457196533, −3.63507821602921171877871336480, −2.93334866743795456422373880043, −0.44945373501547614742774698264, 1.11037056208000841333666017356, 4.85313163604371770114396938655, 6.08653549622527327041965861419, 7.58235856384679552281332154887, 9.067797723810360548913048300662, 12.41942406133403748047664915829, 13.53243831052699625460905600762, 15.89973769098455684359516315677, 16.74292493160179350203421406954, 18.00027365019473063284724159916

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