Properties

Label 2-3-3.2-c40-0-9
Degree $2$
Conductor $3$
Sign $-0.109 + 0.993i$
Analytic cond. $30.4026$
Root an. cond. $5.51386$
Motivic weight $40$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.89e5i·2-s + (−3.83e8 + 3.46e9i)3-s + 8.59e11·4-s − 1.65e14i·5-s + (1.69e15 + 1.87e14i)6-s + 1.22e17·7-s − 9.59e17i·8-s + (−1.18e19 − 2.65e18i)9-s − 8.08e19·10-s + 8.75e19i·11-s + (−3.29e20 + 2.97e21i)12-s − 2.33e22·13-s − 5.98e22i·14-s + (5.72e23 + 6.32e22i)15-s + 4.74e23·16-s − 8.49e23i·17-s + ⋯
L(s)  = 1  − 0.467i·2-s + (−0.109 + 0.993i)3-s + 0.781·4-s − 1.73i·5-s + (0.464 + 0.0513i)6-s + 1.53·7-s − 0.832i·8-s + (−0.975 − 0.218i)9-s − 0.808·10-s + 0.130i·11-s + (−0.0859 + 0.776i)12-s − 1.22·13-s − 0.715i·14-s + (1.72 + 0.190i)15-s + 0.392·16-s − 0.208i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.109 + 0.993i$
Analytic conductor: \(30.4026\)
Root analytic conductor: \(5.51386\)
Motivic weight: \(40\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :20),\ -0.109 + 0.993i)\)

Particular Values

\(L(\frac{41}{2})\) \(\approx\) \(2.360387019\)
\(L(\frac12)\) \(\approx\) \(2.360387019\)
\(L(21)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.83e8 - 3.46e9i)T \)
good2 \( 1 + 4.89e5iT - 1.09e12T^{2} \)
5 \( 1 + 1.65e14iT - 9.09e27T^{2} \)
7 \( 1 - 1.22e17T + 6.36e33T^{2} \)
11 \( 1 - 8.75e19iT - 4.52e41T^{2} \)
13 \( 1 + 2.33e22T + 3.61e44T^{2} \)
17 \( 1 + 8.49e23iT - 1.65e49T^{2} \)
19 \( 1 - 1.15e25T + 1.41e51T^{2} \)
23 \( 1 + 1.67e27iT - 2.94e54T^{2} \)
29 \( 1 + 9.66e28iT - 3.13e58T^{2} \)
31 \( 1 + 6.99e29T + 4.51e59T^{2} \)
37 \( 1 - 2.98e31T + 5.34e62T^{2} \)
41 \( 1 - 1.54e31iT - 3.24e64T^{2} \)
43 \( 1 - 2.10e32T + 2.18e65T^{2} \)
47 \( 1 - 8.11e32iT - 7.65e66T^{2} \)
53 \( 1 + 2.65e34iT - 9.35e68T^{2} \)
59 \( 1 + 3.47e35iT - 6.82e70T^{2} \)
61 \( 1 + 4.26e34T + 2.58e71T^{2} \)
67 \( 1 - 3.93e36T + 1.10e73T^{2} \)
71 \( 1 + 1.43e35iT - 1.12e74T^{2} \)
73 \( 1 + 1.95e36T + 3.41e74T^{2} \)
79 \( 1 + 3.82e37T + 8.03e75T^{2} \)
83 \( 1 - 6.29e37iT - 5.79e76T^{2} \)
89 \( 1 - 1.43e39iT - 9.45e77T^{2} \)
97 \( 1 - 2.20e39T + 2.95e79T^{2} \)
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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

−16.24517802127104154237689047401, −14.74602807496284943092790500867, −12.34855715679214759022341226318, −11.25603763877253484078634906347, −9.581880109869228056601744355674, −8.042296079017148407216919794929, −5.27511849653019745121790702883, −4.35203924634372053119583964015, −2.14544161488662727445885083400, −0.71321626587522863612268031498, 1.74479410079259146522761274017, 2.74408361210093665507283437907, 5.64963573566609416760285196881, 7.13165139620911539456915656262, 7.72924595642653373802161413175, 10.88277781025861537628370866082, 11.71853428215234506787744309575, 14.27145631348598002353672954517, 14.90835752232367392516429119905, 17.29809375475320648216160955583

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