Properties

Label 2-462-33.32-c1-0-21
Degree $2$
Conductor $462$
Sign $0.0705 + 0.997i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (0.482 − 1.66i)3-s + 4-s − 0.885i·5-s + (0.482 − 1.66i)6-s i·7-s + 8-s + (−2.53 − 1.60i)9-s − 0.885i·10-s + (−1.14 − 3.11i)11-s + (0.482 − 1.66i)12-s + 1.32i·13-s i·14-s + (−1.47 − 0.426i)15-s + 16-s + 1.35·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.278 − 0.960i)3-s + 0.5·4-s − 0.395i·5-s + (0.196 − 0.679i)6-s − 0.377i·7-s + 0.353·8-s + (−0.845 − 0.534i)9-s − 0.279i·10-s + (−0.345 − 0.938i)11-s + (0.139 − 0.480i)12-s + 0.368i·13-s − 0.267i·14-s + (−0.380 − 0.110i)15-s + 0.250·16-s + 0.329·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0705 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0705 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.0705 + 0.997i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.0705 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62373 - 1.51290i\)
\(L(\frac12)\) \(\approx\) \(1.62373 - 1.51290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.482 + 1.66i)T \)
7 \( 1 + iT \)
11 \( 1 + (1.14 + 3.11i)T \)
good5 \( 1 + 0.885iT - 5T^{2} \)
13 \( 1 - 1.32iT - 13T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 - 4.67iT - 19T^{2} \)
23 \( 1 + 1.08iT - 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 - 0.592T + 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 - 0.991iT - 43T^{2} \)
47 \( 1 + 0.204iT - 47T^{2} \)
53 \( 1 + 1.07iT - 53T^{2} \)
59 \( 1 + 5.25iT - 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 - 6.28iT - 73T^{2} \)
79 \( 1 + 5.69iT - 79T^{2} \)
83 \( 1 + 9.95T + 83T^{2} \)
89 \( 1 - 6.72iT - 89T^{2} \)
97 \( 1 + 3.90T + 97T^{2} \)
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

−11.10327418223184207634399973307, −10.07707265441890784061231750972, −8.702222050846856895410847189069, −8.073075790196821816182052194786, −7.02789113688931291222164695872, −6.16214172904011766634168665304, −5.21607166352621020903235382883, −3.82815381911416350046807047522, −2.71679137826691718320936204508, −1.17959187559396655041864071780, 2.43453743897452011011300686215, 3.31636658711371993059426576388, 4.62294650114551487269675444137, 5.24299816470281865079068700441, 6.48365970905091706268982572933, 7.55156784540708680365551727203, 8.634406931219916645685494022532, 9.650383277293898305798120035472, 10.45103003049974821926285192707, 11.16238556183187092243283244686

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