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Properties

Label 2-3-3.2-c26-0-2
Degree $2$
Conductor $3$
Sign $0.990 + 0.136i$
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  + 1.54e4i·2-s + (−1.57e6 − 2.17e5i)3-s − 1.71e8·4-s + 7.56e8i·5-s + (3.36e9 − 2.43e10i)6-s − 6.43e10·7-s − 1.60e12i·8-s + (2.44e12 + 6.88e11i)9-s − 1.16e13·10-s + 4.53e13i·11-s + (2.70e14 + 3.72e13i)12-s − 1.22e14·13-s − 9.93e14i·14-s + (1.64e14 − 1.19e15i)15-s + 1.32e16·16-s − 8.44e14i·17-s + ⋯
L(s)  = 1  + 1.88i·2-s + (−0.990 − 0.136i)3-s − 2.54·4-s + 0.619i·5-s + (0.257 − 1.86i)6-s − 0.664·7-s − 2.91i·8-s + (0.962 + 0.270i)9-s − 1.16·10-s + 1.31i·11-s + (2.52 + 0.348i)12-s − 0.403·13-s − 1.25i·14-s + (0.0846 − 0.613i)15-s + 2.95·16-s − 0.0852i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.990 + 0.136i$
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: \((2,\ 3,\ (\ :13),\ 0.990 + 0.136i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.0560596 - 0.00384860i\)
\(L(\frac12)\) \(\approx\) \(0.0560596 - 0.00384860i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57e6 + 2.17e5i)T \)
good2 \( 1 - 1.54e4iT - 6.71e7T^{2} \)
5 \( 1 - 7.56e8iT - 1.49e18T^{2} \)
7 \( 1 + 6.43e10T + 9.38e21T^{2} \)
11 \( 1 - 4.53e13iT - 1.19e27T^{2} \)
13 \( 1 + 1.22e14T + 9.17e28T^{2} \)
17 \( 1 + 8.44e14iT - 9.81e31T^{2} \)
19 \( 1 - 1.38e16T + 1.76e33T^{2} \)
23 \( 1 + 6.13e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.18e19iT - 1.05e38T^{2} \)
31 \( 1 + 2.12e19T + 5.96e38T^{2} \)
37 \( 1 - 2.40e19T + 5.93e40T^{2} \)
41 \( 1 - 2.62e20iT - 8.55e41T^{2} \)
43 \( 1 + 8.46e20T + 2.95e42T^{2} \)
47 \( 1 - 8.66e21iT - 2.98e43T^{2} \)
53 \( 1 + 3.15e22iT - 6.77e44T^{2} \)
59 \( 1 + 4.46e22iT - 1.10e46T^{2} \)
61 \( 1 + 8.62e22T + 2.62e46T^{2} \)
67 \( 1 + 7.14e23T + 3.00e47T^{2} \)
71 \( 1 - 1.40e24iT - 1.35e48T^{2} \)
73 \( 1 - 1.49e24T + 2.79e48T^{2} \)
79 \( 1 + 1.64e24T + 2.17e49T^{2} \)
83 \( 1 + 1.43e25iT - 7.87e49T^{2} \)
89 \( 1 + 2.88e25iT - 4.83e50T^{2} \)
97 \( 1 - 9.43e25T + 4.52e51T^{2} \)
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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

−18.58766843020638943473808151189, −17.44101971815346962366524372040, −16.15073922528537808126210435870, −14.77972900480804358420779671647, −12.79911191669561437171516497600, −9.843017471889067070255532873455, −7.33121785535560659579137311168, −6.32781897458558100488616526119, −4.66650303962481178345975197405, −0.03255576164682747690388780462, 1.18474917458375451113257686082, 3.48074105301868006326383536119, 5.26425418306947791087007838758, 9.229596500740415875859196204498, 10.75326358813829959799030356778, 12.08224316508038951105049600067, 13.30453630048319129905554809468, 16.66280453420223449508926626926, 18.37488787939014411310905362659, 19.75091754598887064096752602391

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