↑

Properties

Label 2-720-60.59-c1-0-6
Degree $2$
Conductor $720$
Sign $0.250 + 0.968i$
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  + (−1.73 + 1.41i)5-s − 4.24·7-s + 2.44·11-s − 2.44i·13-s + 6.92·17-s − 6.92i·19-s − 6i·23-s + (0.999 − 4.89i)25-s − 2.82i·29-s − 3.46i·31-s + (7.34 − 6i)35-s + 2.44i·37-s + 7.07i·41-s − 8.48·43-s + 10.9·49-s + ⋯
L(s)  = 1  + (−0.774 + 0.632i)5-s − 1.60·7-s + 0.738·11-s − 0.679i·13-s + 1.68·17-s − 1.58i·19-s − 1.25i·23-s + (0.199 − 0.979i)25-s − 0.525i·29-s − 0.622i·31-s + (1.24 − 1.01i)35-s + 0.402i·37-s + 1.10i·41-s − 1.29·43-s + 1.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.250 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668554 - 0.517688i\)
\(L(\frac12)\) \(\approx\) \(0.668554 - 0.517688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 1.41i)T \)
good7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 9.79T + 71T^{2} \)
73 \( 1 + 4.89iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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

−10.11051334103622129256698521030, −9.579666947269591575979795545791, −8.480364074254960066287302606547, −7.51768826820634588987350115701, −6.66616299583952713691949008275, −6.06454800907354121026899373074, −4.59757925613144105902620780351, −3.39002847979506869504552085278, −2.88778834536485631000205573100, −0.48166225715866406438412163783, 1.35791019854649732726761464363, 3.49727725006992297713349159886, 3.69746539392307751751406557293, 5.25756123819143537255558452944, 6.17296675006022184423979274197, 7.13818507270306066942301416584, 7.958718307968415063793394567198, 9.049054744200277141080450596922, 9.620477098078728213926657390338, 10.40957199629164651760217938831

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