↑

Properties

Label 2-3-3.2-c40-0-2
Degree $2$
Conductor $3$
Sign $-0.998 + 0.0468i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.37e6i·2-s + (−3.48e9 + 1.63e8i)3-s − 8.02e11·4-s − 4.92e13i·5-s + (−2.25e14 − 4.80e15i)6-s + 5.39e16·7-s + 4.09e17i·8-s + (1.21e19 − 1.13e18i)9-s + 6.78e19·10-s + 2.19e20i·11-s + (2.79e21 − 1.31e20i)12-s + 5.75e21·13-s + 7.43e22i·14-s + (8.03e21 + 1.71e23i)15-s − 1.44e24·16-s + 6.05e24i·17-s + ⋯
L(s)  = 1  + 1.31i·2-s + (−0.998 + 0.0468i)3-s − 0.729·4-s − 0.516i·5-s + (−0.0615 − 1.31i)6-s + 0.675·7-s + 0.355i·8-s + (0.995 − 0.0935i)9-s + 0.678·10-s + 0.326i·11-s + (0.729 − 0.0341i)12-s + 0.302·13-s + 0.889i·14-s + (0.0241 + 0.515i)15-s − 1.19·16-s + 1.48i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0468i)\, \overline{\Lambda}(41-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+20) \, L(s)\cr =\mathstrut & (-0.998 + 0.0468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.998 + 0.0468i$
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: \((2,\ 3,\ (\ :20),\ -0.998 + 0.0468i)\)

Particular Values

\(L(\frac{41}{2})\) \(\approx\) \(1.212681035\)
\(L(\frac12)\) \(\approx\) \(1.212681035\)
\(L(21)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.48e9 - 1.63e8i)T \)
good2 \( 1 - 1.37e6iT - 1.09e12T^{2} \)
5 \( 1 + 4.92e13iT - 9.09e27T^{2} \)
7 \( 1 - 5.39e16T + 6.36e33T^{2} \)
11 \( 1 - 2.19e20iT - 4.52e41T^{2} \)
13 \( 1 - 5.75e21T + 3.61e44T^{2} \)
17 \( 1 - 6.05e24iT - 1.65e49T^{2} \)
19 \( 1 - 3.89e25T + 1.41e51T^{2} \)
23 \( 1 + 2.93e27iT - 2.94e54T^{2} \)
29 \( 1 - 2.25e29iT - 3.13e58T^{2} \)
31 \( 1 + 5.05e29T + 4.51e59T^{2} \)
37 \( 1 + 3.44e31T + 5.34e62T^{2} \)
41 \( 1 + 2.84e31iT - 3.24e64T^{2} \)
43 \( 1 + 3.37e31T + 2.18e65T^{2} \)
47 \( 1 - 4.50e33iT - 7.65e66T^{2} \)
53 \( 1 - 1.59e34iT - 9.35e68T^{2} \)
59 \( 1 - 4.55e35iT - 6.82e70T^{2} \)
61 \( 1 - 2.97e35T + 2.58e71T^{2} \)
67 \( 1 - 1.49e36T + 1.10e73T^{2} \)
71 \( 1 - 8.37e36iT - 1.12e74T^{2} \)
73 \( 1 + 1.01e37T + 3.41e74T^{2} \)
79 \( 1 + 9.72e37T + 8.03e75T^{2} \)
83 \( 1 - 3.08e38iT - 5.79e76T^{2} \)
89 \( 1 + 4.83e38iT - 9.45e77T^{2} \)
97 \( 1 + 7.78e39T + 2.95e79T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.03944775411497856029016623344, −16.03547121053707462950383910548, −14.59040221389056096247509826107, −12.53409727235584559459425582511, −10.81377974021616489529097878150, −8.549292907073456768180292364668, −7.01397376147128738632711138583, −5.65152781243285868586343610976, −4.57457743740486528663145481688, −1.37897130981789857394562423053, 0.46809136442876418126240435144, 1.70900290079736835364504565246, 3.41454038003436982608380583253, 5.17688354136618910402914973230, 7.11994359039077121876887227027, 9.753065374697378717085025517012, 11.17022932728470706940675988714, 11.77938429081580742972628745243, 13.57068179081357273545225392527, 15.88278418215245181263843830161

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