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Properties

Label 2-3-3.2-c68-0-2
Degree $2$
Conductor $3$
Sign $0.190 - 0.981i$
Analytic cond. $87.8517$
Root an. cond. $9.37292$
Motivic weight $68$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.19e9i·2-s + (3.18e15 − 1.63e16i)3-s + 2.28e20·4-s − 5.93e23i·5-s + (1.34e26 + 2.60e25i)6-s − 6.83e28·7-s + 4.28e30i·8-s + (−2.57e32 − 1.04e32i)9-s + 4.85e33·10-s − 1.91e35i·11-s + (7.25e35 − 3.73e36i)12-s + 3.76e37·13-s − 5.59e38i·14-s + (−9.71e39 − 1.88e39i)15-s + 3.21e40·16-s + 1.23e42i·17-s + ⋯
L(s)  = 1  + 0.476i·2-s + (0.190 − 0.981i)3-s + 0.772·4-s − 1.01i·5-s + (0.468 + 0.0909i)6-s − 1.26·7-s + 0.845i·8-s + (−0.927 − 0.374i)9-s + 0.485·10-s − 0.748i·11-s + (0.147 − 0.758i)12-s + 0.502·13-s − 0.601i·14-s + (−1.00 − 0.194i)15-s + 0.369·16-s + 1.80i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(69-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+34) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.190 - 0.981i$
Analytic conductor: \(87.8517\)
Root analytic conductor: \(9.37292\)
Motivic weight: \(68\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :34),\ 0.190 - 0.981i)\)

Particular Values

\(L(\frac{69}{2})\) \(\approx\) \(1.030006574\)
\(L(\frac12)\) \(\approx\) \(1.030006574\)
\(L(35)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.18e15 + 1.63e16i)T \)
good2 \( 1 - 8.19e9iT - 2.95e20T^{2} \)
5 \( 1 + 5.93e23iT - 3.38e47T^{2} \)
7 \( 1 + 6.83e28T + 2.92e57T^{2} \)
11 \( 1 + 1.91e35iT - 6.52e70T^{2} \)
13 \( 1 - 3.76e37T + 5.59e75T^{2} \)
17 \( 1 - 1.23e42iT - 4.68e83T^{2} \)
19 \( 1 + 5.74e43T + 9.02e86T^{2} \)
23 \( 1 - 1.67e46iT - 3.95e92T^{2} \)
29 \( 1 + 2.38e49iT - 2.77e99T^{2} \)
31 \( 1 + 4.91e50T + 2.58e101T^{2} \)
37 \( 1 - 2.86e53T + 4.34e106T^{2} \)
41 \( 1 - 3.97e54iT - 4.66e109T^{2} \)
43 \( 1 - 4.99e55T + 1.19e111T^{2} \)
47 \( 1 + 2.90e56iT - 5.04e113T^{2} \)
53 \( 1 + 1.15e58iT - 1.78e117T^{2} \)
59 \( 1 + 1.83e59iT - 2.61e120T^{2} \)
61 \( 1 + 1.75e60T + 2.52e121T^{2} \)
67 \( 1 + 8.48e61T + 1.48e124T^{2} \)
71 \( 1 + 6.37e62iT - 7.68e125T^{2} \)
73 \( 1 - 2.74e63T + 5.08e126T^{2} \)
79 \( 1 + 2.32e64T + 1.09e129T^{2} \)
83 \( 1 + 7.50e64iT - 3.14e130T^{2} \)
89 \( 1 - 3.38e66iT - 3.61e132T^{2} \)
97 \( 1 - 2.46e67T + 1.26e135T^{2} \)
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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

−13.16814781324095466284543100112, −12.54099369434122303517136427064, −10.93617907213375216713484582581, −8.881080630571091925080147884201, −7.959088792757040196099304228672, −6.37138123340325572121936805133, −5.93310619832756455184251114253, −3.65457698694833888188913217255, −2.22757934990607326073422582991, −1.08865700563209376046619951384, 0.22112562487480163800322348765, 2.41537636848389809106866058136, 2.98107662330483804189361470708, 4.17571162733933888965005807783, 6.16660716996530149814426582984, 7.13187178224182092331580258946, 9.274395841922745462680771796713, 10.34201457656467343003717667897, 11.11697290887432549425584057986, 12.70685690903269226003596041317

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