↑

Properties

Label 2-56-56.51-c2-0-11
Degree $2$
Conductor $56$
Sign $0.568 + 0.822i$
Analytic cond. $1.52588$
Root an. cond. $1.23526$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 + 0.234i)2-s + (−2.66 − 4.61i)3-s + (3.88 + 0.933i)4-s + (−1.86 − 1.07i)5-s + (−4.21 − 9.79i)6-s + (6.91 + 1.06i)7-s + (7.50 + 2.76i)8-s + (−9.71 + 16.8i)9-s + (−3.45 − 2.58i)10-s + (−2.62 − 4.55i)11-s + (−6.06 − 20.4i)12-s + 21.4i·13-s + (13.4 + 3.73i)14-s + 11.5i·15-s + (14.2 + 7.26i)16-s + (−0.463 − 0.802i)17-s + ⋯
L(s)  = 1  + (0.993 + 0.117i)2-s + (−0.888 − 1.53i)3-s + (0.972 + 0.233i)4-s + (−0.373 − 0.215i)5-s + (−0.701 − 1.63i)6-s + (0.988 + 0.152i)7-s + (0.938 + 0.345i)8-s + (−1.07 + 1.86i)9-s + (−0.345 − 0.258i)10-s + (−0.239 − 0.414i)11-s + (−0.505 − 1.70i)12-s + 1.64i·13-s + (0.963 + 0.267i)14-s + 0.766i·15-s + (0.891 + 0.453i)16-s + (−0.0272 − 0.0472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.39783 - 0.733152i\)
\(L(\frac12)\) \(\approx\) \(1.39783 - 0.733152i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 - 0.234i)T \)
7 \( 1 + (-6.91 - 1.06i)T \)
good3 \( 1 + (2.66 + 4.61i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.86 + 1.07i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.62 + 4.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 + (0.463 + 0.802i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.96 + 5.13i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (7.52 + 4.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 9.42iT - 841T^{2} \)
31 \( 1 + (29.8 - 17.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.0 - 6.40i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 43.1T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + (39.8 + 22.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (64.5 - 37.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (26.8 + 46.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-24.0 - 13.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.2 - 67.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 74.5iT - 5.04e3T^{2} \)
73 \( 1 + (16.8 + 29.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-26.1 - 15.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 72.9T + 6.88e3T^{2} \)
89 \( 1 + (-27.4 + 47.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 53.7T + 9.40e3T^{2} \)
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

−14.39879624982076804769095739080, −13.65943497907110113446238932743, −12.50574603439057159508086991378, −11.66429312392550075811685667117, −11.13704235059600980403898670617, −8.229162021055688474811657242627, −7.17132706025960787038253666811, −6.05302801936573758098215232202, −4.70510394468712506387241346069, −1.87443279338652404671037622039, 3.58295877901671650053557757559, 4.86143457724970698700993183808, 5.73505978620930405930087095537, 7.74115391480495346414868268245, 9.927886489049769856929347913106, 10.87716871934947343046907514111, 11.49236557878628567461681089974, 12.77417675212731000532227035770, 14.52494826701520081682425393526, 15.19124081146892713859773119420

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