Properties

Label 2-3-3.2-c26-0-3
Degree $2$
Conductor $3$
Sign $0.669 - 0.743i$
Analytic cond. $12.8487$
Root an. cond. $3.58452$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.23e3i·2-s + (−1.06e6 + 1.18e6i)3-s + 5.66e7·4-s − 1.50e9i·5-s + (−3.83e9 − 3.45e9i)6-s + 7.23e10·7-s + 4.00e11i·8-s + (−2.65e11 − 2.52e12i)9-s + 4.89e12·10-s + 2.86e13i·11-s + (−6.03e13 + 6.70e13i)12-s + 4.12e14·13-s + 2.34e14i·14-s + (1.78e15 + 1.61e15i)15-s + 2.50e15·16-s + 1.55e16i·17-s + ⋯
L(s)  = 1  + 0.395i·2-s + (−0.669 + 0.743i)3-s + 0.843·4-s − 1.23i·5-s + (−0.293 − 0.264i)6-s + 0.746·7-s + 0.729i·8-s + (−0.104 − 0.994i)9-s + 0.489·10-s + 0.829i·11-s + (−0.564 + 0.626i)12-s + 1.36·13-s + 0.295i·14-s + (0.919 + 0.827i)15-s + 0.555·16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.669 - 0.743i$
Analytic conductor: \(12.8487\)
Root analytic conductor: \(3.58452\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :13),\ 0.669 - 0.743i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(1.83133 + 0.815314i\)
\(L(\frac12)\) \(\approx\) \(1.83133 + 0.815314i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.06e6 - 1.18e6i)T \)
good2 \( 1 - 3.23e3iT - 6.71e7T^{2} \)
5 \( 1 + 1.50e9iT - 1.49e18T^{2} \)
7 \( 1 - 7.23e10T + 9.38e21T^{2} \)
11 \( 1 - 2.86e13iT - 1.19e27T^{2} \)
13 \( 1 - 4.12e14T + 9.17e28T^{2} \)
17 \( 1 - 1.55e16iT - 9.81e31T^{2} \)
19 \( 1 - 2.79e15T + 1.76e33T^{2} \)
23 \( 1 + 6.09e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.18e18iT - 1.05e38T^{2} \)
31 \( 1 - 2.23e19T + 5.96e38T^{2} \)
37 \( 1 - 4.09e19T + 5.93e40T^{2} \)
41 \( 1 - 2.65e20iT - 8.55e41T^{2} \)
43 \( 1 - 2.85e21T + 2.95e42T^{2} \)
47 \( 1 + 6.83e21iT - 2.98e43T^{2} \)
53 \( 1 - 2.17e22iT - 6.77e44T^{2} \)
59 \( 1 + 6.38e22iT - 1.10e46T^{2} \)
61 \( 1 + 2.23e23T + 2.62e46T^{2} \)
67 \( 1 + 7.65e23T + 3.00e47T^{2} \)
71 \( 1 - 4.93e22iT - 1.35e48T^{2} \)
73 \( 1 + 7.06e23T + 2.79e48T^{2} \)
79 \( 1 + 1.47e24T + 2.17e49T^{2} \)
83 \( 1 + 1.16e25iT - 7.87e49T^{2} \)
89 \( 1 - 3.65e25iT - 4.83e50T^{2} \)
97 \( 1 + 6.39e25T + 4.52e51T^{2} \)
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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

−20.40489942451735469203258521431, −17.45620755381850990524005891154, −16.35401679030251675705382960233, −15.09538060215664559572698715427, −12.31164384990307926933152276156, −10.74670485892589818784796262171, −8.461990884987792859947104781562, −6.03046591967320993810593986270, −4.45828959463856622344919456871, −1.34902957718148635092211673827, 1.18886224149357199186355610936, 2.88136891017011943692273706029, 6.07892902685884282552543652511, 7.45932540334304089712755043000, 10.90389254070411662292348920129, 11.54085066262804802414621604003, 13.83872155708236988279285460864, 15.94936004495331710423894807663, 18.02052028254202547269944003405, 19.10534311084448684632591068213

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