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Properties

Label 2-3-3.2-c70-0-18
Degree $2$
Conductor $3$
Sign $-0.768 + 0.640i$
Analytic cond. $93.0951$
Root an. cond. $9.64858$
Motivic weight $70$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.54e10i·2-s + (3.84e16 − 3.20e16i)3-s + 9.40e20·4-s + 2.49e24i·5-s + (−4.95e26 − 5.95e26i)6-s − 4.12e29·7-s − 3.28e31i·8-s + (4.51e32 − 2.46e33i)9-s + 3.86e34·10-s − 2.08e36i·11-s + (3.61e37 − 3.01e37i)12-s + 9.30e38·13-s + 6.38e39i·14-s + (7.98e40 + 9.58e40i)15-s + 6.02e41·16-s − 1.16e42i·17-s + ⋯
L(s)  = 1  − 0.450i·2-s + (0.768 − 0.640i)3-s + 0.796·4-s + 0.856i·5-s + (−0.288 − 0.346i)6-s − 1.08·7-s − 0.809i·8-s + (0.180 − 0.983i)9-s + 0.386·10-s − 0.742i·11-s + (0.612 − 0.510i)12-s + 0.956·13-s + 0.490i·14-s + (0.548 + 0.658i)15-s + 0.432·16-s − 0.100i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(71-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35) \, L(s)\cr =\mathstrut & (-0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.768 + 0.640i$
Analytic conductor: \(93.0951\)
Root analytic conductor: \(9.64858\)
Motivic weight: \(70\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :35),\ -0.768 + 0.640i)\)

Particular Values

\(L(\frac{71}{2})\) \(\approx\) \(2.816526397\)
\(L(\frac12)\) \(\approx\) \(2.816526397\)
\(L(36)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.84e16 + 3.20e16i)T \)
good2 \( 1 + 1.54e10iT - 1.18e21T^{2} \)
5 \( 1 - 2.49e24iT - 8.47e48T^{2} \)
7 \( 1 + 4.12e29T + 1.43e59T^{2} \)
11 \( 1 + 2.08e36iT - 7.89e72T^{2} \)
13 \( 1 - 9.30e38T + 9.46e77T^{2} \)
17 \( 1 + 1.16e42iT - 1.35e86T^{2} \)
19 \( 1 + 1.28e44T + 3.25e89T^{2} \)
23 \( 1 + 2.85e47iT - 2.09e95T^{2} \)
29 \( 1 - 6.67e50iT - 2.33e102T^{2} \)
31 \( 1 - 1.17e52T + 2.48e104T^{2} \)
37 \( 1 + 1.02e55T + 5.94e109T^{2} \)
41 \( 1 - 4.80e56iT - 7.85e112T^{2} \)
43 \( 1 + 1.37e57T + 2.20e114T^{2} \)
47 \( 1 + 4.51e58iT - 1.11e117T^{2} \)
53 \( 1 + 1.76e60iT - 5.00e120T^{2} \)
59 \( 1 + 1.74e62iT - 9.11e123T^{2} \)
61 \( 1 + 3.49e62T + 9.39e124T^{2} \)
67 \( 1 - 8.78e63T + 6.68e127T^{2} \)
71 \( 1 + 7.60e64iT - 3.87e129T^{2} \)
73 \( 1 - 2.57e65T + 2.70e130T^{2} \)
79 \( 1 + 1.03e66T + 6.82e132T^{2} \)
83 \( 1 + 2.08e67iT - 2.16e134T^{2} \)
89 \( 1 + 1.88e68iT - 2.86e136T^{2} \)
97 \( 1 + 6.36e69T + 1.18e139T^{2} \)
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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

−12.56009178416311020880685994288, −11.16478478859282718403820488641, −9.951980084360692928549975044683, −8.381417832512089865415785486948, −6.78121253731678341328340833750, −6.36008222034743147632478796580, −3.38915876785451943719235127154, −3.08065625748551956753331991884, −1.79694738609860344723295444214, −0.50696885371000983572052743901, 1.43064040511330527539813570901, 2.72592719544784371647987525884, 3.94944287643675807179054616043, 5.41478966087545434893642363031, 6.81720173637718270357180013641, 8.204429486655663191681370499212, 9.329833398236078237304642926679, 10.60794737659327276885558053746, 12.37551132361976374511362571173, 13.69477124369037276217467505804

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