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Properties

Label 2-720-80.3-c1-0-19
Degree $2$
Conductor $720$
Sign $-0.811 - 0.584i$
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 + i)2-s + 2i·4-s + (2 + i)5-s + (−3 + 3i)7-s + (−2 + 2i)8-s + (1 + 3i)10-s + (1 + i)11-s − 2i·13-s − 6·14-s − 4·16-s + (−1 + i)17-s + (−3 − 3i)19-s + (−2 + 4i)20-s + 2i·22-s + (1 + i)23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (0.894 + 0.447i)5-s + (−1.13 + 1.13i)7-s + (−0.707 + 0.707i)8-s + (0.316 + 0.948i)10-s + (0.301 + 0.301i)11-s − 0.554i·13-s − 1.60·14-s − 16-s + (−0.242 + 0.242i)17-s + (−0.688 − 0.688i)19-s + (−0.447 + 0.894i)20-s + 0.426i·22-s + (0.208 + 0.208i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.811 - 0.584i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.811 - 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629753 + 1.95077i\)
\(L(\frac12)\) \(\approx\) \(0.629753 + 1.95077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (-2 - i)T \)
good7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + (-1 - i)T + 23iT^{2} \)
29 \( 1 + (-7 + 7i)T - 29iT^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-7 - 7i)T + 47iT^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (3 - 3i)T - 59iT^{2} \)
61 \( 1 + (1 + i)T + 61iT^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 + 3i)T - 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (11 - 11i)T - 97iT^{2} \)
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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

−10.73057355101041417394668619248, −9.683766580070126929823902463290, −9.050481160880803499857328237920, −8.124663599516232198582418581196, −6.80141375905686787510881406975, −6.31661779780174955120104631988, −5.64003006683396553648944949424, −4.53594051909741375411514032048, −3.08769866890418410376408389428, −2.45291577844336670654215150193, 0.841826057761572822179262196729, 2.24877343960286584302682143519, 3.54720363056628255249617564685, 4.34955945817994985974837715151, 5.48985654364290840828238081237, 6.44017374849058707932241328492, 6.99116277238255980010032444270, 8.769495901785871104081806800855, 9.394046380951027785864914227502, 10.41260568178774126558927420989

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