Properties

Label 2-56-56.51-c2-0-2
Degree $2$
Conductor $56$
Sign $-0.427 - 0.904i$
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  + (0.104 + 1.99i)2-s + (1.99 + 3.44i)3-s + (−3.97 + 0.418i)4-s + (−1.63 − 0.941i)5-s + (−6.67 + 4.33i)6-s + (5.14 − 4.74i)7-s + (−1.25 − 7.90i)8-s + (−3.42 + 5.93i)9-s + (1.70 − 3.35i)10-s + (3.93 + 6.82i)11-s + (−9.36 − 12.8i)12-s + 11.4i·13-s + (10.0 + 9.78i)14-s − 7.49i·15-s + (15.6 − 3.33i)16-s + (1.44 + 2.51i)17-s + ⋯
L(s)  = 1  + (0.0523 + 0.998i)2-s + (0.663 + 1.14i)3-s + (−0.994 + 0.104i)4-s + (−0.326 − 0.188i)5-s + (−1.11 + 0.722i)6-s + (0.735 − 0.677i)7-s + (−0.156 − 0.987i)8-s + (−0.380 + 0.659i)9-s + (0.170 − 0.335i)10-s + (0.358 + 0.620i)11-s + (−0.780 − 1.07i)12-s + 0.883i·13-s + (0.715 + 0.698i)14-s − 0.499i·15-s + (0.978 − 0.208i)16-s + (0.0852 + 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.427 - 0.904i$
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: \((2,\ 56,\ (\ :1),\ -0.427 - 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.702727 + 1.10956i\)
\(L(\frac12)\) \(\approx\) \(0.702727 + 1.10956i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 - 1.99i)T \)
7 \( 1 + (-5.14 + 4.74i)T \)
good3 \( 1 + (-1.99 - 3.44i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.63 + 0.941i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.93 - 6.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 11.4iT - 169T^{2} \)
17 \( 1 + (-1.44 - 2.51i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-15.0 + 26.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (33.3 + 19.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 27.8iT - 841T^{2} \)
31 \( 1 + (-19.4 + 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (39.4 + 22.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 40.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 + (-71.5 - 41.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (23.2 - 13.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-5.20 - 9.01i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (19.1 + 11.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.6 - 51.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 38.2iT - 5.04e3T^{2} \)
73 \( 1 + (6.98 + 12.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-44.3 - 25.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 89.4T + 6.88e3T^{2} \)
89 \( 1 + (52.6 - 91.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 55.3T + 9.40e3T^{2} \)
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

−15.44839584230233329844298107223, −14.38580160473849630349926715793, −13.84259282084164629269498276925, −12.04198521078933936890216401938, −10.34545589559053418352606145003, −9.278285457201749227345309523924, −8.267454074031609632434947496925, −6.93107727961184744063061256435, −4.77780631456221429226358207675, −4.01135491768329047239550089079, 1.69724132096286795167093314884, 3.36893935798325436592436979866, 5.62429889979023604734862529621, 7.81796336445902143997636043901, 8.458605401488312039013013915772, 10.06193654856841545451259051947, 11.67203267756941129135468001019, 12.18177352300606465246083024515, 13.56688396876946603708783653091, 14.14522891867154621164051786619

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