↑

Properties

Label 2-84e2-28.27-c1-0-41
Degree $2$
Conductor $7056$
Sign $-0.101 - 0.994i$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  + 3.21i·5-s − 1.36i·11-s + 2.93i·13-s + 6.91i·17-s + 7.35·19-s − 3.62i·23-s − 5.33·25-s + 1.11·29-s + 8.70·31-s + 7.63·37-s − 0.833i·41-s − 4.82i·43-s − 2.95·47-s + 4.57·53-s + 4.39·55-s + ⋯
L(s)  = 1  + 1.43i·5-s − 0.412i·11-s + 0.812i·13-s + 1.67i·17-s + 1.68·19-s − 0.756i·23-s − 1.06·25-s + 0.206·29-s + 1.56·31-s + 1.25·37-s − 0.130i·41-s − 0.735i·43-s − 0.430·47-s + 0.628·53-s + 0.592·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.210219217\)
\(L(\frac12)\) \(\approx\) \(2.210219217\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.21iT - 5T^{2} \)
11 \( 1 + 1.36iT - 11T^{2} \)
13 \( 1 - 2.93iT - 13T^{2} \)
17 \( 1 - 6.91iT - 17T^{2} \)
19 \( 1 - 7.35T + 19T^{2} \)
23 \( 1 + 3.62iT - 23T^{2} \)
29 \( 1 - 1.11T + 29T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 - 7.63T + 37T^{2} \)
41 \( 1 + 0.833iT - 41T^{2} \)
43 \( 1 + 4.82iT - 43T^{2} \)
47 \( 1 + 2.95T + 47T^{2} \)
53 \( 1 - 4.57T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 - 6.49iT - 73T^{2} \)
79 \( 1 + 7.79iT - 79T^{2} \)
83 \( 1 + 8.87T + 83T^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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

−8.193225700408318562366592177658, −7.23793425581885885349382166591, −6.77245140310074892147786727547, −6.14265814080574739412678389456, −5.51070640334171377144443880045, −4.37093803211219393715643167666, −3.71519467565717389394924328198, −2.93099879418250677302641874321, −2.25012989504867609492030004418, −1.06047251964743364901601701237, 0.68872666997758125210417531654, 1.20034081092970956479154574615, 2.55641745632363535304130445731, 3.28299654484618887006305156828, 4.39075223549704732732998388464, 5.00741635101150358621443652759, 5.37394928053242055351301119568, 6.25225638606545702655521338343, 7.31968271241587675780290838579, 7.72164977954857806990913913545

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