Properties

Degree $2$
Conductor $7056$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 6·11-s + 6·13-s − 2·17-s + 4·19-s + 2·23-s − 25-s − 8·29-s + 4·31-s − 6·37-s + 10·41-s + 4·43-s + 4·47-s + 4·53-s + 12·55-s + 12·59-s + 2·61-s + 12·65-s − 12·67-s + 6·71-s + 2·73-s + 8·79-s − 4·85-s − 14·89-s + 8·95-s + 2·97-s − 18·101-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.80·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.417·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s − 0.986·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 0.549·53-s + 1.61·55-s + 1.56·59-s + 0.256·61-s + 1.48·65-s − 1.46·67-s + 0.712·71-s + 0.234·73-s + 0.900·79-s − 0.433·85-s − 1.48·89-s + 0.820·95-s + 0.203·97-s − 1.79·101-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: $1$
Motivic weight: \(1\)
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.282992578\)
\(L(\frac12)\) \(\approx\) \(3.282992578\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−7.998025692462453871157050453993, −7.03117849106627501533503077601, −6.52183409924032017598847199663, −5.84508098578725880263863151392, −5.37484584785383850457030776799, −3.99575772917284247763386451724, −3.86709904214332908635948878848, −2.66547321765012652301986475968, −1.60524385357388864055460274101, −1.05060246088758382568109167441, 1.05060246088758382568109167441, 1.60524385357388864055460274101, 2.66547321765012652301986475968, 3.86709904214332908635948878848, 3.99575772917284247763386451724, 5.37484584785383850457030776799, 5.84508098578725880263863151392, 6.52183409924032017598847199663, 7.03117849106627501533503077601, 7.998025692462453871157050453993

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