↑

Properties

Label 2-720-80.27-c1-0-23
Degree $2$
Conductor $720$
Sign $0.971 - 0.237i$
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  + (1.23 − 0.687i)2-s + (1.05 − 1.69i)4-s + (−0.832 + 2.07i)5-s + (2.83 + 2.83i)7-s + (0.134 − 2.82i)8-s + (0.399 + 3.13i)10-s + (−1.95 + 1.95i)11-s + 2.05i·13-s + (5.45 + 1.55i)14-s + (−1.77 − 3.58i)16-s + (4.06 + 4.06i)17-s + (−0.683 + 0.683i)19-s + (2.65 + 3.60i)20-s + (−1.07 + 3.76i)22-s + (4.95 − 4.95i)23-s + ⋯
L(s)  = 1  + (0.873 − 0.486i)2-s + (0.527 − 0.849i)4-s + (−0.372 + 0.928i)5-s + (1.07 + 1.07i)7-s + (0.0473 − 0.998i)8-s + (0.126 + 0.992i)10-s + (−0.590 + 0.590i)11-s + 0.569i·13-s + (1.45 + 0.415i)14-s + (−0.444 − 0.895i)16-s + (0.986 + 0.986i)17-s + (−0.156 + 0.156i)19-s + (0.592 + 0.805i)20-s + (−0.228 + 0.803i)22-s + (1.03 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55992 + 0.308684i\)
\(L(\frac12)\) \(\approx\) \(2.55992 + 0.308684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 + 0.687i)T \)
3 \( 1 \)
5 \( 1 + (0.832 - 2.07i)T \)
good7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 - 1.95i)T - 11iT^{2} \)
13 \( 1 - 2.05iT - 13T^{2} \)
17 \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \)
19 \( 1 + (0.683 - 0.683i)T - 19iT^{2} \)
23 \( 1 + (-4.95 + 4.95i)T - 23iT^{2} \)
29 \( 1 + (0.835 + 0.835i)T + 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 + 4.54iT - 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 - 0.849iT - 43T^{2} \)
47 \( 1 + (-2.72 + 2.72i)T - 47iT^{2} \)
53 \( 1 + 5.17T + 53T^{2} \)
59 \( 1 + (-4.16 - 4.16i)T + 59iT^{2} \)
61 \( 1 + (-5.55 + 5.55i)T - 61iT^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (4.39 + 4.39i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 2.75T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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

−10.75843765351299718139683965053, −9.918584490162507580709978529055, −8.710027883287416228255769099369, −7.72482338846953705427435802121, −6.78765305715138343549539836691, −5.78210668722076224363933984315, −4.95947477924944136034368004847, −3.93079899544055460823829231494, −2.71924991682344631732368469011, −1.85881385036400464265293690147, 1.14073892554907882156992967011, 3.04253537564476226001364530849, 4.09802029391201533137782107029, 5.06785213326432718353735740289, 5.46707600057315379161568705403, 7.00743179469520723748555581219, 7.86027463867319887389888983000, 8.132449423238914594350399350754, 9.391528477958750029153001356631, 10.69176550272062425457985508618

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