↑

Properties

Label 2-720-80.69-c1-0-11
Degree $2$
Conductor $720$
Sign $0.986 - 0.161i$
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  + (−1.39 + 0.258i)2-s + (1.86 − 0.719i)4-s + (−2.19 − 0.404i)5-s − 1.81·7-s + (−2.40 + 1.48i)8-s + (3.16 − 0.00742i)10-s + (0.331 + 0.331i)11-s + (0.0310 + 0.0310i)13-s + (2.52 − 0.469i)14-s + (2.96 − 2.68i)16-s + 1.00i·17-s + (−2.08 + 2.08i)19-s + (−4.39 + 0.828i)20-s + (−0.547 − 0.375i)22-s + 6.22·23-s + ⋯
L(s)  = 1  + (−0.983 + 0.183i)2-s + (0.933 − 0.359i)4-s + (−0.983 − 0.180i)5-s − 0.686·7-s + (−0.851 + 0.524i)8-s + (0.999 − 0.00234i)10-s + (0.100 + 0.100i)11-s + (0.00860 + 0.00860i)13-s + (0.674 − 0.125i)14-s + (0.740 − 0.671i)16-s + 0.243i·17-s + (−0.477 + 0.477i)19-s + (−0.982 + 0.185i)20-s + (−0.116 − 0.0800i)22-s + 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.986 - 0.161i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694438 + 0.0563147i\)
\(L(\frac12)\) \(\approx\) \(0.694438 + 0.0563147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (2.19 + 0.404i)T \)
good7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 + (-0.331 - 0.331i)T + 11iT^{2} \)
13 \( 1 + (-0.0310 - 0.0310i)T + 13iT^{2} \)
17 \( 1 - 1.00iT - 17T^{2} \)
19 \( 1 + (2.08 - 2.08i)T - 19iT^{2} \)
23 \( 1 - 6.22T + 23T^{2} \)
29 \( 1 + (-6.28 + 6.28i)T - 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (-0.0723 + 0.0723i)T - 37iT^{2} \)
41 \( 1 - 3.06iT - 41T^{2} \)
43 \( 1 + (3.78 - 3.78i)T - 43iT^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (-7.04 + 7.04i)T - 53iT^{2} \)
59 \( 1 + (-6.68 - 6.68i)T + 59iT^{2} \)
61 \( 1 + (-2.89 + 2.89i)T - 61iT^{2} \)
67 \( 1 + (-0.150 - 0.150i)T + 67iT^{2} \)
71 \( 1 - 14.5iT - 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (-5.48 - 5.48i)T + 83iT^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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

−10.24518091043525899498265085325, −9.611652622972356001252202043038, −8.479685682366884088796551272757, −8.137885714667196585900240670786, −6.93729239079875551788548985609, −6.43886293840811718672341983214, −5.08240145627882333675848181185, −3.77123793156016913322447323819, −2.61526100523465630492781657737, −0.816156749821622410473461126377, 0.78481839423319970229399819681, 2.70721539944571431913894124635, 3.49915819164909565185687769465, 4.84749361645760021437676340009, 6.47756164305081780411119901362, 6.92795660953598780310035709602, 7.928876283554670476378899795114, 8.719515003776894978204027878069, 9.418346365740569242198027005527, 10.51164962663077856058607137613

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