Properties

Label 2-84e2-1.1-c1-0-7
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s − 6·13-s + 2·17-s − 4·19-s + 8·23-s − 25-s + 2·29-s − 10·37-s − 6·41-s + 4·43-s − 6·53-s + 8·55-s − 4·59-s − 6·61-s + 12·65-s − 4·67-s + 8·71-s − 10·73-s + 4·83-s − 4·85-s − 6·89-s + 8·95-s + 14·97-s − 2·101-s + 8·103-s + 12·107-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s − 1.64·37-s − 0.937·41-s + 0.609·43-s − 0.824·53-s + 1.07·55-s − 0.520·59-s − 0.768·61-s + 1.48·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.439·83-s − 0.433·85-s − 0.635·89-s + 0.820·95-s + 1.42·97-s − 0.199·101-s + 0.788·103-s + 1.16·107-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: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6882534522\)
\(L(\frac12)\) \(\approx\) \(0.6882534522\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−7.80482566859023040657170709303, −7.35990014451951924962892403121, −6.78096076788874774298742993601, −5.71112834827052213288460713965, −4.89765783876193378617287021019, −4.61184535669995648177321109940, −3.42407012452672903343907771151, −2.84079499225778567119125782528, −1.91012367697335827208229040334, −0.39903622293453192344515596766, 0.39903622293453192344515596766, 1.91012367697335827208229040334, 2.84079499225778567119125782528, 3.42407012452672903343907771151, 4.61184535669995648177321109940, 4.89765783876193378617287021019, 5.71112834827052213288460713965, 6.78096076788874774298742993601, 7.35990014451951924962892403121, 7.80482566859023040657170709303

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