↑

Properties

Label 2-720-20.3-c1-0-11
Degree $2$
Conductor $720$
Sign $0.525 + 0.850i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (1 − 2i)5-s + (1 + i)13-s + (5 − 5i)17-s + (−3 − 4i)25-s − 10i·29-s + (−7 + 7i)37-s + 10·41-s − 7i·49-s + (−5 − 5i)53-s + 12·61-s + (3 − i)65-s + (11 + 11i)73-s + (−5 − 15i)85-s + 10i·89-s + (−13 + 13i)97-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + (0.277 + 0.277i)13-s + (1.21 − 1.21i)17-s + (−0.600 − 0.800i)25-s − 1.85i·29-s + (−1.15 + 1.15i)37-s + 1.56·41-s − i·49-s + (−0.686 − 0.686i)53-s + 1.53·61-s + (0.372 − 0.124i)65-s + (1.28 + 1.28i)73-s + (−0.542 − 1.62i)85-s + 1.05i·89-s + (−1.31 + 1.31i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42303 - 0.793392i\)
\(L(\frac12)\) \(\approx\) \(1.42303 - 0.793392i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 10iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (7 - 7i)T - 37iT^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11 - 11i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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

−9.898988187197046592924298763821, −9.629131391752696114180756406965, −8.538180891490929448922770681746, −7.82855208493149221467800522786, −6.69874058099085780973968331031, −5.66122178961127731544996289487, −4.93054464557888470245271706652, −3.80709863824822138295433709877, −2.38768127530993581969491773566, −0.935267726284866606307417625882, 1.58826915175872574550972198792, 2.98570042786697898155467027824, 3.85724090966648733626557431643, 5.37953566580412955310819941787, 6.06599255800950452550590219230, 7.05712736452429743387604982721, 7.86095739511536240670876666551, 8.892777616076392053113395387391, 9.813040406290643222927654857079, 10.66662421035005122413665592620

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