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Properties

Label 2-720-240.179-c1-0-29
Degree $2$
Conductor $720$
Sign $-0.221 + 0.975i$
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  + (−0.612 − 1.27i)2-s + (−1.24 + 1.56i)4-s + (−0.263 − 2.22i)5-s + 2.95i·7-s + (2.75 + 0.636i)8-s + (−2.66 + 1.69i)10-s + (1.04 − 1.04i)11-s + (0.00780 − 0.00780i)13-s + (3.77 − 1.81i)14-s + (−0.875 − 3.90i)16-s − 2.00·17-s + (5.10 − 5.10i)19-s + (3.79 + 2.36i)20-s + (−1.96 − 0.688i)22-s + 6.49·23-s + ⋯
L(s)  = 1  + (−0.433 − 0.901i)2-s + (−0.624 + 0.780i)4-s + (−0.117 − 0.993i)5-s + 1.11i·7-s + (0.974 + 0.225i)8-s + (−0.844 + 0.536i)10-s + (0.313 − 0.313i)11-s + (0.00216 − 0.00216i)13-s + (1.00 − 0.484i)14-s + (−0.218 − 0.975i)16-s − 0.485·17-s + (1.17 − 1.17i)19-s + (0.848 + 0.528i)20-s + (−0.418 − 0.146i)22-s + 1.35·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.680654 - 0.852899i\)
\(L(\frac12)\) \(\approx\) \(0.680654 - 0.852899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.612 + 1.27i)T \)
3 \( 1 \)
5 \( 1 + (0.263 + 2.22i)T \)
good7 \( 1 - 2.95iT - 7T^{2} \)
11 \( 1 + (-1.04 + 1.04i)T - 11iT^{2} \)
13 \( 1 + (-0.00780 + 0.00780i)T - 13iT^{2} \)
17 \( 1 + 2.00T + 17T^{2} \)
19 \( 1 + (-5.10 + 5.10i)T - 19iT^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + (-5.45 + 5.45i)T - 29iT^{2} \)
31 \( 1 + 4.99iT - 31T^{2} \)
37 \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + (3.52 - 3.52i)T - 43iT^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + (-2.98 + 2.98i)T - 53iT^{2} \)
59 \( 1 + (-5.47 + 5.47i)T - 59iT^{2} \)
61 \( 1 + (4.67 + 4.67i)T + 61iT^{2} \)
67 \( 1 + (2.29 + 2.29i)T + 67iT^{2} \)
71 \( 1 - 0.212iT - 71T^{2} \)
73 \( 1 - 16.4T + 73T^{2} \)
79 \( 1 + 4.54iT - 79T^{2} \)
83 \( 1 + (4.68 - 4.68i)T - 83iT^{2} \)
89 \( 1 - 0.123T + 89T^{2} \)
97 \( 1 - 8.29iT - 97T^{2} \)
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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

−9.953953106456123810515165085024, −9.274354911581092016766584301986, −8.689345156802027225135887439167, −8.022877662509821414935262086300, −6.72750662574933737079515635040, −5.30532296965731268591908882671, −4.68527406503862592674273565296, −3.33789772760387343774239931832, −2.23997434570157725744649512084, −0.78467768461189580750741099608, 1.28208927560046576498013301070, 3.25654812967734110621608451667, 4.32672426818713431677370510995, 5.43288755337329015886504160552, 6.67049198950356597934916734542, 7.06461820498725398506959438710, 7.81315244449281943507082055208, 8.860201853670146452707623920936, 9.875425284609141354113028882273, 10.45883370376569490601822961812

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