↑

Properties

Label 2-840-280.139-c1-0-35
Degree $2$
Conductor $840$
Sign $-0.618 - 0.785i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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  + (0.716 + 1.21i)2-s + 3-s + (−0.973 + 1.74i)4-s + (−0.151 + 2.23i)5-s + (0.716 + 1.21i)6-s + (2.00 − 1.72i)7-s + (−2.82 + 0.0635i)8-s + 9-s + (−2.82 + 1.41i)10-s + 2.25·11-s + (−0.973 + 1.74i)12-s + 6.09i·13-s + (3.54 + 1.20i)14-s + (−0.151 + 2.23i)15-s + (−2.10 − 3.40i)16-s − 2.64·17-s + ⋯
L(s)  = 1  + (0.506 + 0.862i)2-s + 0.577·3-s + (−0.486 + 0.873i)4-s + (−0.0678 + 0.997i)5-s + (0.292 + 0.497i)6-s + (0.756 − 0.653i)7-s + (−0.999 + 0.0224i)8-s + 0.333·9-s + (−0.894 + 0.446i)10-s + 0.680·11-s + (−0.281 + 0.504i)12-s + 1.68i·13-s + (0.946 + 0.321i)14-s + (−0.0391 + 0.576i)15-s + (−0.525 − 0.850i)16-s − 0.641·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.618 - 0.785i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05274 + 2.16808i\)
\(L(\frac12)\) \(\approx\) \(1.05274 + 2.16808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.716 - 1.21i)T \)
3 \( 1 - T \)
5 \( 1 + (0.151 - 2.23i)T \)
7 \( 1 + (-2.00 + 1.72i)T \)
good11 \( 1 - 2.25T + 11T^{2} \)
13 \( 1 - 6.09iT - 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 1.31iT - 19T^{2} \)
23 \( 1 - 6.62T + 23T^{2} \)
29 \( 1 - 6.44iT - 29T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + 2.30T + 37T^{2} \)
41 \( 1 + 5.50iT - 41T^{2} \)
43 \( 1 + 1.26iT - 43T^{2} \)
47 \( 1 + 0.599iT - 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 0.847iT - 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 - 3.94iT - 71T^{2} \)
73 \( 1 - 8.34T + 73T^{2} \)
79 \( 1 + 0.674iT - 79T^{2} \)
83 \( 1 - 7.23T + 83T^{2} \)
89 \( 1 - 5.66iT - 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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

−10.64099477420817599432063605519, −9.154076467211295617893464026006, −8.920131810079930759465152243982, −7.61521173123535262469638728198, −6.98043497007197008971843124930, −6.58691255184129233063217710277, −5.07765833908889876876566396370, −4.15224197236277951145832139299, −3.42956496403389468422242031564, −1.99859249517482144320764669542, 1.02211794053103778064429880822, 2.19484313529399639146666036629, 3.35252614829711586169746712084, 4.43650027176564038095924093222, 5.21851559035348439052310863736, 6.00544638548431309182229748097, 7.59643824630069933051001537519, 8.565489090368646366278815916141, 9.002348515268535811223413279858, 9.876702145803033026441702745449

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