↑

Properties

Label 2-56-56.11-c4-0-22
Degree $2$
Conductor $56$
Sign $0.691 + 0.722i$
Analytic cond. $5.78871$
Root an. cond. $2.40597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.229 + 3.99i)2-s + (7.74 − 13.4i)3-s + (−15.8 − 1.83i)4-s + (−13.5 + 7.81i)5-s + (51.7 + 34.0i)6-s + (43.5 − 22.4i)7-s + (10.9 − 63.0i)8-s + (−79.4 − 137. i)9-s + (−28.0 − 55.8i)10-s + (81.1 − 140. i)11-s + (−147. + 198. i)12-s − 187. i·13-s + (79.6 + 179. i)14-s + 241. i·15-s + (249. + 58.3i)16-s + (−128. + 222. i)17-s + ⋯
L(s)  = 1  + (−0.0574 + 0.998i)2-s + (0.860 − 1.49i)3-s + (−0.993 − 0.114i)4-s + (−0.541 + 0.312i)5-s + (1.43 + 0.944i)6-s + (0.888 − 0.458i)7-s + (0.171 − 0.985i)8-s + (−0.980 − 1.69i)9-s + (−0.280 − 0.558i)10-s + (0.670 − 1.16i)11-s + (−1.02 + 1.38i)12-s − 1.10i·13-s + (0.406 + 0.913i)14-s + 1.07i·15-s + (0.973 + 0.228i)16-s + (−0.444 + 0.769i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(5.78871\)
Root analytic conductor: \(2.40597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :2),\ 0.691 + 0.722i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.55797 - 0.665350i\)
\(L(\frac12)\) \(\approx\) \(1.55797 - 0.665350i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.229 - 3.99i)T \)
7 \( 1 + (-43.5 + 22.4i)T \)
good3 \( 1 + (-7.74 + 13.4i)T + (-40.5 - 70.1i)T^{2} \)
5 \( 1 + (13.5 - 7.81i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-81.1 + 140. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 187. iT - 2.85e4T^{2} \)
17 \( 1 + (128. - 222. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-97.3 - 168. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (7.27 - 4.19i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 1.45e3iT - 7.07e5T^{2} \)
31 \( 1 + (330. + 190. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-2.29e3 + 1.32e3i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 434.T + 2.82e6T^{2} \)
43 \( 1 - 291.T + 3.41e6T^{2} \)
47 \( 1 + (93.1 - 53.7i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.74e3 - 1.00e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (1.00e3 - 1.74e3i)T + (-6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.83e3 - 1.63e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.39e3 + 4.15e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 5.28e3iT - 2.54e7T^{2} \)
73 \( 1 + (-1.14e3 + 1.97e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (2.59e3 - 1.49e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.31e4T + 4.74e7T^{2} \)
89 \( 1 + (2.62e3 + 4.54e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 1.62e4T + 8.85e7T^{2} \)
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

−14.40101009257934737322961357858, −13.53306064608099101778267129477, −12.55253808105973251839659658881, −11.00006848809609239155750152560, −8.849011565252282249169725911799, −7.989791655897774423412900178494, −7.29634796770581545948424024727, −5.93541480346191078322824226073, −3.59146025570420670190894664482, −1.03760436389308722313072965798, 2.32281946910325517256304020242, 4.21038817906317688226705322778, 4.68837959642791898556534458780, 8.027600421790773724764363646480, 9.156792849850621432197041651313, 9.730652536574107171782450260031, 11.25503540762152828433354989782, 11.93932612228594654460759370181, 13.73782234433827970724083259427, 14.65057652088043689517815821924

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