↑

Properties

Label 2-560-7.4-c1-0-11
Degree 22
Conductor 560560
Sign −0.386+0.922i-0.386 + 0.922i
Analytic cond. 4.471624.47162
Root an. cond. 2.114622.11462
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 2.59i)7-s + (−3 − 5.19i)9-s + (−1 + 1.73i)11-s − 6·13-s − 3·15-s + (−1 + 1.73i)17-s + (7.5 + 2.59i)21-s + (−4.5 − 7.79i)23-s + (−0.499 + 0.866i)25-s + 9·27-s + 3·29-s + (1 − 1.73i)31-s + (−3 − 5.19i)33-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (0.223 + 0.387i)5-s + (−0.188 − 0.981i)7-s + (−1 − 1.73i)9-s + (−0.301 + 0.522i)11-s − 1.66·13-s − 0.774·15-s + (−0.242 + 0.420i)17-s + (1.63 + 0.566i)21-s + (−0.938 − 1.62i)23-s + (−0.0999 + 0.173i)25-s + 1.73·27-s + 0.557·29-s + (0.179 − 0.311i)31-s + (−0.522 − 0.904i)33-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=((−0.386+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+1/2) L(s)=((−0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: −0.386+0.922i-0.386 + 0.922i
Analytic conductor: 4.471624.47162
Root analytic conductor: 2.114622.11462
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ560(81,⋅)\chi_{560} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 560, ( :1/2), −0.386+0.922i)(2,\ 560,\ (\ :1/2),\ -0.386 + 0.922i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(0.5+2.59i)T 1 + (0.5 + 2.59i)T
good3 1+(1.5−2.59i)T+(−1.5−2.59i)T2 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2}
11 1+(1−1.73i)T+(−5.5−9.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1+6T+13T2 1 + 6T + 13T^{2}
17 1+(1−1.73i)T+(−8.5−14.7i)T2 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2}
19 1+(−9.5+16.4i)T2 1 + (-9.5 + 16.4i)T^{2}
23 1+(4.5+7.79i)T+(−11.5+19.9i)T2 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1+(−1+1.73i)T+(−15.5−26.8i)T2 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2}
37 1+(4+6.92i)T+(−18.5+32.0i)T2 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2}
41 1−5T+41T2 1 - 5T + 41T^{2}
43 1+T+43T2 1 + T + 43T^{2}
47 1+(−4−6.92i)T+(−23.5+40.7i)T2 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2}
53 1+(2−3.46i)T+(−26.5−45.8i)T2 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2}
59 1+(4−6.92i)T+(−29.5−51.0i)T2 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.5+6.06i)T+(−30.5+52.8i)T2 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.5−2.59i)T+(−33.5−58.0i)T2 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+(7−12.1i)T+(−36.5−63.2i)T2 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2−3.46i)T+(−39.5+68.4i)T2 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2}
83 1−T+83T2 1 - T + 83T^{2}
89 1+(6.5+11.2i)T+(−44.5+77.0i)T2 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2}
97 1+10T+97T2 1 + 10T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.32405943704433599832285826342, −10.03491122470901708091516793093, −9.135668869186383838277426400648, −7.69300552879846633339242529603, −6.71468216381318829784125468996, −5.73764375565985845850705092204, −4.58956822920757238304778436227, −4.17734235668272847511171272854, −2.67102895762323323698297291756, 0, 1.73440781798954448186915495075, 2.78382669842538044614324828656, 4.98578943868135646803071325141, 5.59833468637058787611191790817, 6.45656769547870055658996455581, 7.40348490250964578414996655636, 8.141731613205317063466886155332, 9.229950720918807290995336527717, 10.19928638796643000909459275249

Graph of the ZZ-function along the critical line