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Properties

Label 2-720-80.43-c1-0-22
Degree $2$
Conductor $720$
Sign $0.623 - 0.782i$
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  + (−1.40 − 0.137i)2-s + (1.96 + 0.387i)4-s + (1.37 + 1.76i)5-s + (0.159 − 0.159i)7-s + (−2.70 − 0.814i)8-s + (−1.69 − 2.66i)10-s + (−1.60 + 1.60i)11-s + 4.36·13-s + (−0.246 + 0.202i)14-s + (3.70 + 1.51i)16-s + (4.63 − 4.63i)17-s + (−3.97 + 3.97i)19-s + (2.01 + 3.99i)20-s + (2.48 − 2.04i)22-s + (5.58 + 5.58i)23-s + ⋯
L(s)  = 1  + (−0.995 − 0.0972i)2-s + (0.981 + 0.193i)4-s + (0.615 + 0.787i)5-s + (0.0602 − 0.0602i)7-s + (−0.957 − 0.288i)8-s + (−0.536 − 0.844i)10-s + (−0.485 + 0.485i)11-s + 1.21·13-s + (−0.0658 + 0.0541i)14-s + (0.925 + 0.379i)16-s + (1.12 − 1.12i)17-s + (−0.912 + 0.912i)19-s + (0.451 + 0.892i)20-s + (0.529 − 0.435i)22-s + (1.16 + 1.16i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.623 - 0.782i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.623 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995343 + 0.479687i\)
\(L(\frac12)\) \(\approx\) \(0.995343 + 0.479687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.137i)T \)
3 \( 1 \)
5 \( 1 + (-1.37 - 1.76i)T \)
good7 \( 1 + (-0.159 + 0.159i)T - 7iT^{2} \)
11 \( 1 + (1.60 - 1.60i)T - 11iT^{2} \)
13 \( 1 - 4.36T + 13T^{2} \)
17 \( 1 + (-4.63 + 4.63i)T - 17iT^{2} \)
19 \( 1 + (3.97 - 3.97i)T - 19iT^{2} \)
23 \( 1 + (-5.58 - 5.58i)T + 23iT^{2} \)
29 \( 1 + (6.25 + 6.25i)T + 29iT^{2} \)
31 \( 1 - 1.69iT - 31T^{2} \)
37 \( 1 - 0.609T + 37T^{2} \)
41 \( 1 + 0.538iT - 41T^{2} \)
43 \( 1 + 0.592T + 43T^{2} \)
47 \( 1 + (-4.85 - 4.85i)T + 47iT^{2} \)
53 \( 1 - 4.82iT - 53T^{2} \)
59 \( 1 + (-5.78 - 5.78i)T + 59iT^{2} \)
61 \( 1 + (-1.65 + 1.65i)T - 61iT^{2} \)
67 \( 1 + 0.485T + 67T^{2} \)
71 \( 1 + 6.86T + 71T^{2} \)
73 \( 1 + (-0.160 + 0.160i)T - 73iT^{2} \)
79 \( 1 - 7.13T + 79T^{2} \)
83 \( 1 - 6.88iT - 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (-9.64 + 9.64i)T - 97iT^{2} \)
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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

−10.47003216361272328976465873333, −9.678174081910711844865348894501, −9.037888464800382991300451648301, −7.80718450503246435943575713388, −7.31438204485680731514391997077, −6.21666842900035416371457233265, −5.51433222995789304239385395511, −3.66483568881548957167320855377, −2.64712681162791011721050206749, −1.41007195381467408008206028703, 0.881572556667917232585149088211, 2.10996680489190555658478714317, 3.52999986605154327056696653024, 5.14485931078948425848763224027, 5.95937702047217417228466971598, 6.77498634651032334430971791011, 8.054622042182705038020263970423, 8.671923837713664115529417185734, 9.161329846035679802632095864400, 10.41724546976843920975389543909

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