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Properties

Label 2-720-9.4-c1-0-23
Degree $2$
Conductor $720$
Sign $-0.974 + 0.226i$
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  + (0.403 − 1.68i)3-s + (0.5 − 0.866i)5-s + (−1.91 − 3.32i)7-s + (−2.67 − 1.35i)9-s + (−0.853 − 1.47i)11-s + (−1.85 + 3.21i)13-s + (−1.25 − 1.19i)15-s + 1.70·17-s − 0.292·19-s + (−6.36 + 1.89i)21-s + (−2.91 + 5.05i)23-s + (−0.499 − 0.866i)25-s + (−3.36 + 3.95i)27-s + (−4.33 − 7.50i)29-s + (0.146 − 0.253i)31-s + ⋯
L(s)  = 1  + (0.232 − 0.972i)3-s + (0.223 − 0.387i)5-s + (−0.724 − 1.25i)7-s + (−0.891 − 0.452i)9-s + (−0.257 − 0.445i)11-s + (−0.514 + 0.890i)13-s + (−0.324 − 0.307i)15-s + 0.414·17-s − 0.0671·19-s + (−1.38 + 0.412i)21-s + (−0.608 + 1.05i)23-s + (−0.0999 − 0.173i)25-s + (−0.648 + 0.761i)27-s + (−0.804 − 1.39i)29-s + (0.0262 − 0.0455i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.974 + 0.226i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120095 - 1.04872i\)
\(L(\frac12)\) \(\approx\) \(0.120095 - 1.04872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.403 + 1.68i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1.91 + 3.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.853 + 1.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.85 - 3.21i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 0.292T + 19T^{2} \)
23 \( 1 + (2.91 - 5.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.33 + 7.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.146 + 0.253i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.9T + 37T^{2} \)
41 \( 1 + (-3.48 + 6.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.85 - 3.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.936 - 1.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-5.83 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.48 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.77 + 8.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.91 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-1.83 - 3.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.706299376687960450871856620233, −9.373227484989073502836022147808, −7.938716547495270265597322853542, −7.55654529561757668709375913204, −6.50301817141661218293585970575, −5.81486889187424915719897980138, −4.35110216595338596530501191004, −3.30753049816362679075775244399, −1.93110333746865282281134164469, −0.50478112418109654823917691098, 2.49643176388229704555432561515, 3.06200929740441950232620394382, 4.42244351938899880994863349546, 5.51959248971281751479417490919, 6.07412410176417887978200952559, 7.43651157287797361436857774408, 8.432095509979150015310090197643, 9.264462930737879738023933940893, 9.922385521675488365229239444764, 10.54663052783952631091237525988

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