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Properties

Label 2-3-3.2-c40-0-0
Degree $2$
Conductor $3$
Sign $0.441 + 0.897i$
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  + 1.47e6i·2-s + (1.53e9 + 3.12e9i)3-s − 1.06e12·4-s + 3.18e13i·5-s + (−4.59e15 + 2.26e15i)6-s + 7.85e14·7-s + 5.54e16i·8-s + (−7.42e18 + 9.62e18i)9-s − 4.68e19·10-s − 8.83e20i·11-s + (−1.63e21 − 3.32e21i)12-s − 1.50e22·13-s + 1.15e21i·14-s + (−9.96e22 + 4.90e22i)15-s − 1.24e24·16-s − 2.97e24i·17-s + ⋯
L(s)  = 1  + 1.40i·2-s + (0.441 + 0.897i)3-s − 0.965·4-s + 0.334i·5-s + (−1.25 + 0.618i)6-s + 0.00984·7-s + 0.0481i·8-s + (−0.610 + 0.792i)9-s − 0.468·10-s − 1.31i·11-s + (−0.426 − 0.866i)12-s − 0.790·13-s + 0.0138i·14-s + (−0.299 + 0.147i)15-s − 1.03·16-s − 0.731i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.441 + 0.897i$
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: \((2,\ 3,\ (\ :20),\ 0.441 + 0.897i)\)

Particular Values

\(L(\frac{41}{2})\) \(\approx\) \(0.7467176774\)
\(L(\frac12)\) \(\approx\) \(0.7467176774\)
\(L(21)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53e9 - 3.12e9i)T \)
good2 \( 1 - 1.47e6iT - 1.09e12T^{2} \)
5 \( 1 - 3.18e13iT - 9.09e27T^{2} \)
7 \( 1 - 7.85e14T + 6.36e33T^{2} \)
11 \( 1 + 8.83e20iT - 4.52e41T^{2} \)
13 \( 1 + 1.50e22T + 3.61e44T^{2} \)
17 \( 1 + 2.97e24iT - 1.65e49T^{2} \)
19 \( 1 + 6.09e25T + 1.41e51T^{2} \)
23 \( 1 - 1.99e27iT - 2.94e54T^{2} \)
29 \( 1 + 8.61e28iT - 3.13e58T^{2} \)
31 \( 1 - 3.82e29T + 4.51e59T^{2} \)
37 \( 1 + 1.30e31T + 5.34e62T^{2} \)
41 \( 1 - 1.97e32iT - 3.24e64T^{2} \)
43 \( 1 - 8.12e32T + 2.18e65T^{2} \)
47 \( 1 - 4.06e33iT - 7.65e66T^{2} \)
53 \( 1 - 2.48e34iT - 9.35e68T^{2} \)
59 \( 1 + 1.16e35iT - 6.82e70T^{2} \)
61 \( 1 + 5.58e35T + 2.58e71T^{2} \)
67 \( 1 + 4.40e36T + 1.10e73T^{2} \)
71 \( 1 + 1.87e37iT - 1.12e74T^{2} \)
73 \( 1 - 8.39e35T + 3.41e74T^{2} \)
79 \( 1 + 8.62e37T + 8.03e75T^{2} \)
83 \( 1 - 1.72e38iT - 5.79e76T^{2} \)
89 \( 1 - 3.16e38iT - 9.45e77T^{2} \)
97 \( 1 - 2.49e39T + 2.95e79T^{2} \)
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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

−17.20115150036749164785222400207, −16.07927856793179322877211615431, −14.91245898042768487500271496794, −13.85002504945370013861922907637, −10.99204316217815597548179937084, −9.109528933335420734816092912808, −7.81294030312935102802396802778, −6.09260018676517099869190132132, −4.65738107928460773562436549237, −2.81492108555081140016646017508, 0.20484845774615889342933477327, 1.67396585384565468210286630756, 2.54433618912655427589325292427, 4.32351760188592927963836217858, 6.88094756005686890136363568224, 8.778222338214610660093953108297, 10.39076764928919118637389983776, 12.28999289859407486170481539942, 12.81519295145502361794459134252, 14.74449438763872395406178369603

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