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Properties

Label 2-3-3.2-c32-0-5
Degree $2$
Conductor $3$
Sign $0.874 + 0.484i$
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  − 5.40e4i·2-s + (3.76e7 + 2.08e7i)3-s + 1.37e9·4-s − 5.87e9i·5-s + (1.12e12 − 2.03e12i)6-s + 1.74e13·7-s − 3.06e14i·8-s + (9.82e14 + 1.57e15i)9-s − 3.17e14·10-s + 7.02e16i·11-s + (5.18e16 + 2.87e16i)12-s + 3.65e17·13-s − 9.43e17i·14-s + (1.22e17 − 2.21e17i)15-s − 1.06e19·16-s − 2.53e19i·17-s + ⋯
L(s)  = 1  − 0.824i·2-s + (0.874 + 0.484i)3-s + 0.320·4-s − 0.0384i·5-s + (0.399 − 0.720i)6-s + 0.525·7-s − 1.08i·8-s + (0.530 + 0.847i)9-s − 0.0317·10-s + 1.52i·11-s + (0.280 + 0.155i)12-s + 0.548·13-s − 0.433i·14-s + (0.0186 − 0.0336i)15-s − 0.576·16-s − 0.520i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.874 + 0.484i$
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: \((2,\ 3,\ (\ :16),\ 0.874 + 0.484i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(3.17266 - 0.820089i\)
\(L(\frac12)\) \(\approx\) \(3.17266 - 0.820089i\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.76e7 - 2.08e7i)T \)
good2 \( 1 + 5.40e4iT - 4.29e9T^{2} \)
5 \( 1 + 5.87e9iT - 2.32e22T^{2} \)
7 \( 1 - 1.74e13T + 1.10e27T^{2} \)
11 \( 1 - 7.02e16iT - 2.11e33T^{2} \)
13 \( 1 - 3.65e17T + 4.42e35T^{2} \)
17 \( 1 + 2.53e19iT - 2.36e39T^{2} \)
19 \( 1 - 3.08e20T + 8.31e40T^{2} \)
23 \( 1 + 8.16e21iT - 3.76e43T^{2} \)
29 \( 1 - 4.10e22iT - 6.26e46T^{2} \)
31 \( 1 + 1.05e24T + 5.29e47T^{2} \)
37 \( 1 - 1.22e25T + 1.52e50T^{2} \)
41 \( 1 + 4.93e25iT - 4.06e51T^{2} \)
43 \( 1 - 8.98e25T + 1.86e52T^{2} \)
47 \( 1 - 9.37e26iT - 3.21e53T^{2} \)
53 \( 1 - 3.10e27iT - 1.50e55T^{2} \)
59 \( 1 + 3.29e28iT - 4.64e56T^{2} \)
61 \( 1 + 5.17e28T + 1.35e57T^{2} \)
67 \( 1 + 1.75e29T + 2.71e58T^{2} \)
71 \( 1 - 2.75e29iT - 1.73e59T^{2} \)
73 \( 1 + 3.51e29T + 4.22e59T^{2} \)
79 \( 1 + 3.55e30T + 5.29e60T^{2} \)
83 \( 1 + 2.83e30iT - 2.57e61T^{2} \)
89 \( 1 - 1.02e31iT - 2.40e62T^{2} \)
97 \( 1 + 5.51e31T + 3.77e63T^{2} \)
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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

−18.44810257111558218929788816406, −16.03339519479151038684834515133, −14.58182528017638196674087541972, −12.63130400560585474336212547288, −10.82840095210650176168697866615, −9.408543791870347036192721707815, −7.37277358954455469633483464754, −4.46506279558335241309862528385, −2.80957839905058613594330131953, −1.51383087727851512689728916636, 1.39007541162590128601111677727, 3.20192629420040226426676520396, 5.85312802807427177208323602555, 7.49685021144936875120874055929, 8.682843588356254344547345323028, 11.33314012796050212772090236676, 13.60575016484529092598769243069, 14.85343124507039347747599007726, 16.33916812373128358912265776142, 18.21806414893010178065458091436

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