Properties

Label 2-84e2-1.1-c1-0-63
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 3·11-s − 2·13-s − 6·17-s + 2·19-s + 6·23-s + 4·25-s + 9·29-s − 7·31-s − 10·37-s + 4·43-s + 12·47-s − 3·53-s − 9·55-s − 3·59-s + 4·61-s + 6·65-s − 2·67-s − 2·73-s − 5·79-s + 9·83-s + 18·85-s + 6·89-s − 6·95-s + 13·97-s − 6·101-s − 16·103-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.904·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.67·29-s − 1.25·31-s − 1.64·37-s + 0.609·43-s + 1.75·47-s − 0.412·53-s − 1.21·55-s − 0.390·59-s + 0.512·61-s + 0.744·65-s − 0.244·67-s − 0.234·73-s − 0.562·79-s + 0.987·83-s + 1.95·85-s + 0.635·89-s − 0.615·95-s + 1.31·97-s − 0.597·101-s − 1.57·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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

−7.42889004200935283095691888550, −7.03499149561317652197237395885, −6.43279889442694837895749028653, −5.30774285622288732019904751538, −4.60628839537154430808649782406, −4.00360016328830772086960715028, −3.28885916258822933348831841519, −2.36884465130644791006072860892, −1.12153207349611637065994537186, 0, 1.12153207349611637065994537186, 2.36884465130644791006072860892, 3.28885916258822933348831841519, 4.00360016328830772086960715028, 4.60628839537154430808649782406, 5.30774285622288732019904751538, 6.43279889442694837895749028653, 7.03499149561317652197237395885, 7.42889004200935283095691888550

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