↑

Properties

Label 2-1080-9.7-c1-0-0
Degree $2$
Conductor $1080$
Sign $-0.996 - 0.0825i$
Analytic cond. $8.62384$
Root an. cond. $2.93663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−1.72 + 2.98i)7-s + 2·17-s − 6.89·19-s + (−3.72 − 6.45i)23-s + (−0.499 + 0.866i)25-s + (−0.949 + 1.64i)29-s + (−0.550 − 0.953i)31-s + 3.44·35-s − 6·37-s + (−4.94 − 8.57i)41-s + (−5.89 + 10.2i)43-s + (−4.72 + 8.18i)47-s + (−2.44 − 4.24i)49-s − 7.79·53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.651 + 1.12i)7-s + 0.485·17-s − 1.58·19-s + (−0.776 − 1.34i)23-s + (−0.0999 + 0.173i)25-s + (−0.176 + 0.305i)29-s + (−0.0988 − 0.171i)31-s + 0.583·35-s − 0.986·37-s + (−0.772 − 1.33i)41-s + (−0.899 + 1.55i)43-s + (−0.689 + 1.19i)47-s + (−0.349 − 0.606i)49-s − 1.07·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.996 - 0.0825i$
Analytic conductor: \(8.62384\)
Root analytic conductor: \(2.93663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1080,\ (\ :1/2),\ -0.996 - 0.0825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1436763461\)
\(L(\frac12)\) \(\approx\) \(0.1436763461\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1.72 - 2.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
23 \( 1 + (3.72 + 6.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.949 - 1.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.550 + 0.953i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (4.94 + 8.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.89 - 10.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.79T + 53T^{2} \)
59 \( 1 + (0.550 + 0.953i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.62 - 11.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (-3.44 + 5.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.72 - 4.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.79T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.19794261469485419541695826539, −9.382594719130161025226890167805, −8.581531421442103070293117972514, −8.087122211417057906801208505941, −6.73138337424949266241157975719, −6.11955539230789093602824617567, −5.16230639874031826229770460040, −4.16654922699815692333691593001, −3.01733042177730785109238896502, −1.93330422536802253890014374955, 0.05979139259002987583721154762, 1.83343945285897842594516457148, 3.38480871880939296899959248258, 3.91045233589265619263917539167, 5.11337154680617306866542993463, 6.31577908838859483460534078229, 6.90738084165821278728894796759, 7.76077103819260610863682643708, 8.554828269778591653399234291802, 9.749430213815210710513578338762

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