Properties

Label 2-84e2-12.11-c1-0-33
Degree $2$
Conductor $7056$
Sign $0.995 - 0.0917i$
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  − 2.73i·5-s − 4.24·11-s + 6.31·13-s − 0.732i·17-s + 6.69i·19-s + 9.14·23-s − 2.46·25-s − 2.44i·29-s + 1.79i·31-s − 3.46·37-s + 4.19i·41-s + 9.46i·43-s + 3.46·47-s − 1.41i·53-s + 11.5i·55-s + ⋯
L(s)  = 1  − 1.22i·5-s − 1.27·11-s + 1.75·13-s − 0.177i·17-s + 1.53i·19-s + 1.90·23-s − 0.492·25-s − 0.454i·29-s + 0.322i·31-s − 0.569·37-s + 0.655i·41-s + 1.44i·43-s + 0.505·47-s − 0.194i·53-s + 1.56i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.995 - 0.0917i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (4607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 0.995 - 0.0917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.957240419\)
\(L(\frac12)\) \(\approx\) \(1.957240419\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73iT - 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 - 6.31T + 13T^{2} \)
17 \( 1 + 0.732iT - 17T^{2} \)
19 \( 1 - 6.69iT - 19T^{2} \)
23 \( 1 - 9.14T + 23T^{2} \)
29 \( 1 + 2.44iT - 29T^{2} \)
31 \( 1 - 1.79iT - 31T^{2} \)
37 \( 1 + 3.46T + 37T^{2} \)
41 \( 1 - 4.19iT - 41T^{2} \)
43 \( 1 - 9.46iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 1.41iT - 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 8.10T + 61T^{2} \)
67 \( 1 - 12.9iT - 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 8.10T + 73T^{2} \)
79 \( 1 + 8.53iT - 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 - 9.66iT - 89T^{2} \)
97 \( 1 + 4.52T + 97T^{2} \)
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

−8.074455861832779624461330221917, −7.44880900547427016606311017336, −6.40083784291737324882969513129, −5.74657068116037721194024652107, −5.13874484095021815344870826436, −4.48909870170676376654665817881, −3.59128299671166531610959586264, −2.82411115687996380536055747182, −1.53064303693446841352460777451, −0.922984408085791023709192296811, 0.59728318824190725357734001025, 1.92304012806657679482380016743, 3.01067195711220464809800509353, 3.17718444287993614108502521582, 4.31067142380059836498252206572, 5.22060526238058816731317464594, 5.82648904891043142441985472945, 6.76185443775894178765141313267, 7.05712120404434002659193476877, 7.80600883113204285897495974897

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