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Properties

Label 2-720-15.8-c1-0-2
Degree $2$
Conductor $720$
Sign $-0.374 - 0.927i$
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  + (0.707 + 2.12i)5-s + (−2 + 2i)7-s − 2.82i·11-s + (3 + 3i)13-s + (−1.41 − 1.41i)17-s + 4i·19-s + (−5.65 + 5.65i)23-s + (−3.99 + 3i)25-s − 9.89·29-s + 8·31-s + (−5.65 − 2.82i)35-s + (−3 + 3i)37-s − 1.41i·41-s + (8.48 + 8.48i)47-s − i·49-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)5-s + (−0.755 + 0.755i)7-s − 0.852i·11-s + (0.832 + 0.832i)13-s + (−0.342 − 0.342i)17-s + 0.917i·19-s + (−1.17 + 1.17i)23-s + (−0.799 + 0.600i)25-s − 1.83·29-s + 1.43·31-s + (−0.956 − 0.478i)35-s + (−0.493 + 0.493i)37-s − 0.220i·41-s + (1.23 + 1.23i)47-s − 0.142i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.660246 + 0.978521i\)
\(L(\frac12)\) \(\approx\) \(0.660246 + 0.978521i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 2.12i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (5.65 - 5.65i)T - 23iT^{2} \)
29 \( 1 + 9.89T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-8.48 - 8.48i)T + 47iT^{2} \)
53 \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + (8 - 8i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
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

−10.68682439115387352423460166421, −9.742374550570625353844875447199, −9.130504304216525929211584731319, −8.118610472669692296181108858854, −7.05977632213055431396055662264, −6.03178310616356836964693964274, −5.79413736096819793170551158263, −3.94579918076548362111975035057, −3.13488813947697232335197629858, −1.91820950984907582639690913729, 0.59829248653955591051148185110, 2.17944852982505820453673977806, 3.74495328496290865116212101767, 4.54940342328145707829952932644, 5.69862461239992987827074594328, 6.55562903589419399218114980432, 7.56753713705238076935838029161, 8.513647468645948387704149723777, 9.300519115297908930438498242872, 10.16974024552876129119546476722

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