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Properties

Label 2-720-20.3-c1-0-8
Degree $2$
Conductor $720$
Sign $0.850 - 0.525i$
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  + (2 + i)5-s + (2.82 + 2.82i)7-s − 5.65i·11-s + (3 + 3i)13-s + (1 − i)17-s − 5.65·19-s + (−2.82 + 2.82i)23-s + (3 + 4i)25-s + 4i·29-s + (2.82 + 8.48i)35-s + (5 − 5i)37-s + (2.82 − 2.82i)43-s + (−2.82 − 2.82i)47-s + 9.00i·49-s + (−1 − i)53-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)5-s + (1.06 + 1.06i)7-s − 1.70i·11-s + (0.832 + 0.832i)13-s + (0.242 − 0.242i)17-s − 1.29·19-s + (−0.589 + 0.589i)23-s + (0.600 + 0.800i)25-s + 0.742i·29-s + (0.478 + 1.43i)35-s + (0.821 − 0.821i)37-s + (0.431 − 0.431i)43-s + (−0.412 − 0.412i)47-s + 1.28i·49-s + (−0.137 − 0.137i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91378 + 0.543667i\)
\(L(\frac12)\) \(\approx\) \(1.91378 + 0.543667i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2 - i)T \)
good7 \( 1 + (-2.82 - 2.82i)T + 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-2.82 + 2.82i)T - 43iT^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + (2.82 + 2.82i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 5.65T + 79T^{2} \)
83 \( 1 + (2.82 - 2.82i)T - 83iT^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−10.76207403052580283443672966675, −9.474359041355659881832898766857, −8.695240151836359690834611604879, −8.225371976256451933934288823100, −6.77492387315936618083378343711, −5.87663439525393893898199069515, −5.41394752360444055875605633043, −3.94410361057665538280631075859, −2.63852224880232550203143512485, −1.59861957531367978836704335989, 1.25108832525556024564387860561, 2.27886909060435536651965762010, 4.17912950374236370212943521181, 4.68808491534124084129503227756, 5.86288057028163082038441799367, 6.81047690892648088793871774872, 7.88675247607973060769692218302, 8.448617636884204419795370340990, 9.711849828816361933576352486788, 10.29084235082692176748229223671

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