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Properties

Label 2-720-80.69-c1-0-16
Degree $2$
Conductor $720$
Sign $0.522 - 0.852i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 0.563i)2-s + (1.36 − 1.46i)4-s + (2.18 + 0.496i)5-s + 0.862·7-s + (−0.949 + 2.66i)8-s + (−3.10 + 0.583i)10-s + (4.00 + 4.00i)11-s + (−1.17 − 1.17i)13-s + (−1.11 + 0.485i)14-s + (−0.267 − 3.99i)16-s + 3.66i·17-s + (−0.732 + 0.732i)19-s + (3.70 − 2.50i)20-s + (−7.44 − 2.93i)22-s − 4.97·23-s + ⋯
L(s)  = 1  + (−0.917 + 0.398i)2-s + (0.683 − 0.730i)4-s + (0.975 + 0.221i)5-s + 0.325·7-s + (−0.335 + 0.941i)8-s + (−0.982 + 0.184i)10-s + (1.20 + 1.20i)11-s + (−0.326 − 0.326i)13-s + (−0.298 + 0.129i)14-s + (−0.0669 − 0.997i)16-s + 0.888i·17-s + (−0.167 + 0.167i)19-s + (0.828 − 0.560i)20-s + (−1.58 − 0.626i)22-s − 1.03·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.522 - 0.852i$
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: \((2,\ 720,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10957 + 0.621608i\)
\(L(\frac12)\) \(\approx\) \(1.10957 + 0.621608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.563i)T \)
3 \( 1 \)
5 \( 1 + (-2.18 - 0.496i)T \)
good7 \( 1 - 0.862T + 7T^{2} \)
11 \( 1 + (-4.00 - 4.00i)T + 11iT^{2} \)
13 \( 1 + (1.17 + 1.17i)T + 13iT^{2} \)
17 \( 1 - 3.66iT - 17T^{2} \)
19 \( 1 + (0.732 - 0.732i)T - 19iT^{2} \)
23 \( 1 + 4.97T + 23T^{2} \)
29 \( 1 + (-0.253 + 0.253i)T - 29iT^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 + (-5.48 + 5.48i)T - 37iT^{2} \)
41 \( 1 + 9.27iT - 41T^{2} \)
43 \( 1 + (2.43 - 2.43i)T - 43iT^{2} \)
47 \( 1 - 6.79iT - 47T^{2} \)
53 \( 1 + (-2.86 + 2.86i)T - 53iT^{2} \)
59 \( 1 + (-3.30 - 3.30i)T + 59iT^{2} \)
61 \( 1 + (3.44 - 3.44i)T - 61iT^{2} \)
67 \( 1 + (-8.23 - 8.23i)T + 67iT^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 0.0873T + 79T^{2} \)
83 \( 1 + (7.17 + 7.17i)T + 83iT^{2} \)
89 \( 1 + 5.86iT - 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.17142835923936472239044833020, −9.769330544984946988036330682400, −8.935941258689289447305353109191, −8.003615184525112457301752911849, −7.03847588735072023862290340061, −6.30866929618930936685062409987, −5.48569783416502284392745164792, −4.20387122289263256256246584276, −2.35924465259203173619270304027, −1.48591562055812201778573019288, 1.00011322644039807101735468566, 2.19697158760873932847590067245, 3.40346776783431756853721854241, 4.76278412353314993349133356837, 6.16222565755911037407467319759, 6.66878031634385042250731596473, 7.988951379079733074196454683269, 8.703473722789618638907321974186, 9.506928652430681113132276220578, 10.00046995298285424628261159445

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