Properties

Label 2-720-9.7-c1-0-17
Degree $2$
Conductor $720$
Sign $-0.939 + 0.342i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (1.5 − 2.59i)9-s + (2 + 3.46i)13-s + (1.5 + 0.866i)15-s − 6·17-s − 2·19-s − 1.73i·21-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s + 5.19i·27-s + (−1.5 + 2.59i)29-s + (−5 − 8.66i)31-s + 0.999·35-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.188 + 0.327i)7-s + (0.5 − 0.866i)9-s + (0.554 + 0.960i)13-s + (0.387 + 0.223i)15-s − 1.45·17-s − 0.458·19-s − 0.377i·21-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s + 0.999i·27-s + (−0.278 + 0.482i)29-s + (−0.898 − 1.55i)31-s + 0.169·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.12004617562558317612234555629, −9.049173408442582028060569306632, −8.693431316048271361538352350262, −7.15346157054947000600534800765, −6.41203781582680632548827353130, −5.51434012765184138238758748848, −4.47202894275194399058100555398, −3.77857782832053877074108713568, −1.94778818271643003958566757316, 0, 1.73870141948584193105546101954, 3.28228619378010980214582817528, 4.49613787151169548431890054640, 5.55427569378275538933118585294, 6.47477624670373675122207936470, 7.13070494244411027727927907447, 8.050711853716119625023850986512, 9.012758654883809749389358977690, 10.41940890496104677867537904419

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