↑

Properties

Label 2-700-28.11-c0-0-1
Degree 22
Conductor 700700
Sign −0.553+0.832i-0.553 + 0.832i
Analytic cond. 0.3493450.349345
Root an. cond. 0.5910540.591054
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.866 + 0.499i)12-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.499 + 0.866i)24-s + i·27-s + (−0.866 + 0.499i)28-s + 29-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.866 + 0.499i)12-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 + 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.499 + 0.866i)24-s + i·27-s + (−0.866 + 0.499i)28-s + 29-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.553+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s) L(s)=((−0.553+0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.553+0.832i-0.553 + 0.832i
Analytic conductor: 0.3493450.349345
Root analytic conductor: 0.5910540.591054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ700(151,⋅)\chi_{700} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :0), −0.553+0.832i)(2,\ 700,\ (\ :0),\ -0.553 + 0.832i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0248685941.024868594
L(12)L(\frac12) ≈\approx 1.0248685941.024868594
L(1)L(1) 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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
5 1 1
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good3 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
29 1−T+T2 1 - T + T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
41 1+T+T2 1 + T + T^{2}
43 1−iT−T2 1 - iT - T^{2}
47 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
53 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1−iT−T2 1 - iT - T^{2}
89 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+T2 1 + T^{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

−10.57394624317188050831943098573, −9.883294561219959206858428489273, −8.802159915662516793696487388050, −7.20237275803542727261896525076, −6.64445824387371851371414029105, −5.89436365706709778425679089108, −4.94059903892542441235336612623, −3.79940559461868676251154183679, −2.73366311502421252743007216621, −0.984672679369739259079610876653, 2.57013339079330267561694860797, 3.67789542204281055769983056446, 4.82484768556985802310350256888, 5.52760340457769480408736273100, 6.30174284513544197052612291144, 7.10872698690888113528102105175, 8.274464093805326766521557867755, 9.226642579717488817866322626041, 10.31102105306279847053213948223, 11.07473173706261922537963971950

Graph of the ZZ-function along the critical line