Properties

Label 2-3-3.2-c18-0-4
Degree $2$
Conductor $3$
Sign $-0.618 + 0.785i$
Analytic cond. $6.16158$
Root an. cond. $2.48225$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 425. i·2-s + (1.21e4 − 1.54e4i)3-s + 8.11e4·4-s − 7.87e5i·5-s + (−6.58e6 − 5.17e6i)6-s − 3.80e7·7-s − 1.46e8i·8-s + (−9.11e7 − 3.76e8i)9-s − 3.35e8·10-s + 3.08e9i·11-s + (9.87e8 − 1.25e9i)12-s + 1.20e10·13-s + 1.61e10i·14-s + (−1.21e10 − 9.58e9i)15-s − 4.08e10·16-s − 1.44e11i·17-s + ⋯
L(s)  = 1  − 0.830i·2-s + (0.618 − 0.785i)3-s + 0.309·4-s − 0.403i·5-s + (−0.653 − 0.513i)6-s − 0.943·7-s − 1.08i·8-s + (−0.235 − 0.971i)9-s − 0.335·10-s + 1.30i·11-s + (0.191 − 0.243i)12-s + 1.13·13-s + 0.783i·14-s + (−0.316 − 0.249i)15-s − 0.594·16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.618 + 0.785i$
Analytic conductor: \(6.16158\)
Root analytic conductor: \(2.48225\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :9),\ -0.618 + 0.785i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.877817 - 1.80777i\)
\(L(\frac12)\) \(\approx\) \(0.877817 - 1.80777i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.21e4 + 1.54e4i)T \)
good2 \( 1 + 425. iT - 2.62e5T^{2} \)
5 \( 1 + 7.87e5iT - 3.81e12T^{2} \)
7 \( 1 + 3.80e7T + 1.62e15T^{2} \)
11 \( 1 - 3.08e9iT - 5.55e18T^{2} \)
13 \( 1 - 1.20e10T + 1.12e20T^{2} \)
17 \( 1 + 1.44e11iT - 1.40e22T^{2} \)
19 \( 1 - 3.34e11T + 1.04e23T^{2} \)
23 \( 1 - 1.92e12iT - 3.24e24T^{2} \)
29 \( 1 - 1.09e13iT - 2.10e26T^{2} \)
31 \( 1 - 2.95e13T + 6.99e26T^{2} \)
37 \( 1 + 1.11e14T + 1.68e28T^{2} \)
41 \( 1 - 3.22e14iT - 1.07e29T^{2} \)
43 \( 1 + 4.11e13T + 2.52e29T^{2} \)
47 \( 1 + 8.18e14iT - 1.25e30T^{2} \)
53 \( 1 - 7.04e14iT - 1.08e31T^{2} \)
59 \( 1 - 3.09e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.15e16T + 1.36e32T^{2} \)
67 \( 1 + 5.60e15T + 7.40e32T^{2} \)
71 \( 1 - 5.38e16iT - 2.10e33T^{2} \)
73 \( 1 - 1.65e16T + 3.46e33T^{2} \)
79 \( 1 + 1.66e16T + 1.43e34T^{2} \)
83 \( 1 + 4.84e16iT - 3.49e34T^{2} \)
89 \( 1 - 5.03e16iT - 1.22e35T^{2} \)
97 \( 1 + 1.01e18T + 5.77e35T^{2} \)
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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

−20.68733474552617662851320371422, −19.86047568868096861048996241819, −18.34715581113816531011922290515, −15.79915870126307033203670850997, −13.31277110908984203301800330263, −12.01801297980422112153158910080, −9.539596258056412087260220552730, −6.97578087424956301278800764643, −3.10475372478871522884758767004, −1.24154417391559903032098438629, 3.18651791608639209177268081610, 6.15854778481231968025290218798, 8.459791233845743354907479843072, 10.75062418148835970374885015041, 13.91955782710433412103737814269, 15.55867620595009917719742416165, 16.53184342372694803095449548650, 19.18100408014114769722870300107, 20.85529623403621763430530730844, 22.54736376904789266222939608065

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