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Properties

Label 2-3-3.2-c36-0-8
Degree $2$
Conductor $3$
Sign $-0.987 - 0.157i$
Analytic cond. $24.6273$
Root an. cond. $4.96259$
Motivic weight $36$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54e5i·2-s + (−3.82e8 − 6.10e7i)3-s + 4.47e10·4-s + 6.48e12i·5-s + (−9.45e12 + 5.92e13i)6-s − 5.18e14·7-s − 1.75e16i·8-s + (1.42e17 + 4.66e16i)9-s + 1.00e18·10-s − 4.26e18i·11-s + (−1.71e19 − 2.72e18i)12-s − 5.64e19·13-s + 8.03e19i·14-s + (3.95e20 − 2.48e21i)15-s + 3.48e20·16-s + 1.28e22i·17-s + ⋯
L(s)  = 1  − 0.591i·2-s + (−0.987 − 0.157i)3-s + 0.650·4-s + 1.70i·5-s + (−0.0930 + 0.583i)6-s − 0.318·7-s − 0.975i·8-s + (0.950 + 0.311i)9-s + 1.00·10-s − 0.767i·11-s + (−0.642 − 0.102i)12-s − 0.501·13-s + 0.188i·14-s + (0.267 − 1.67i)15-s + 0.0738·16-s + 0.912i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.987 - 0.157i$
Analytic conductor: \(24.6273\)
Root analytic conductor: \(4.96259\)
Motivic weight: \(36\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :18),\ -0.987 - 0.157i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(0.04651046688\)
\(L(\frac12)\) \(\approx\) \(0.04651046688\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.82e8 + 6.10e7i)T \)
good2 \( 1 + 1.54e5iT - 6.87e10T^{2} \)
5 \( 1 - 6.48e12iT - 1.45e25T^{2} \)
7 \( 1 + 5.18e14T + 2.65e30T^{2} \)
11 \( 1 + 4.26e18iT - 3.09e37T^{2} \)
13 \( 1 + 5.64e19T + 1.26e40T^{2} \)
17 \( 1 - 1.28e22iT - 1.97e44T^{2} \)
19 \( 1 + 8.30e22T + 1.08e46T^{2} \)
23 \( 1 + 4.68e24iT - 1.05e49T^{2} \)
29 \( 1 - 3.01e26iT - 4.42e52T^{2} \)
31 \( 1 + 1.30e27T + 4.88e53T^{2} \)
37 \( 1 + 4.79e26T + 2.85e56T^{2} \)
41 \( 1 + 1.03e29iT - 1.14e58T^{2} \)
43 \( 1 + 4.86e27T + 6.38e58T^{2} \)
47 \( 1 + 6.60e29iT - 1.56e60T^{2} \)
53 \( 1 + 9.19e30iT - 1.18e62T^{2} \)
59 \( 1 - 3.49e30iT - 5.63e63T^{2} \)
61 \( 1 + 1.54e32T + 1.87e64T^{2} \)
67 \( 1 + 7.20e32T + 5.47e65T^{2} \)
71 \( 1 + 2.30e33iT - 4.41e66T^{2} \)
73 \( 1 - 3.43e33T + 1.20e67T^{2} \)
79 \( 1 + 1.90e34T + 2.06e68T^{2} \)
83 \( 1 + 8.44e33iT - 1.22e69T^{2} \)
89 \( 1 - 2.14e35iT - 1.50e70T^{2} \)
97 \( 1 - 1.21e35T + 3.34e71T^{2} \)
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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

−16.50287624304604611003760264615, −14.84494802284153088109018917328, −12.55664272224089142161828682488, −10.98058137318336029867489736427, −10.46844762666730946115881545474, −7.06499642490286698644401882209, −6.17043154885672692768342332469, −3.47383673786197691960977962242, −2.02687691957045215412025742080, −0.01643414604380455014021311077, 1.61851187774991455676441379273, 4.63211561198509072678070456405, 5.76274627022596838927266975526, 7.46487757752446142405549370516, 9.523128534244284794557110238484, 11.63286626180479223987944763787, 12.80726514961944677539119393834, 15.49128457816680274199789752645, 16.53790648582538205438921148175, 17.40320743469946221458573022694

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