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Properties

Label 2-2520-280.269-c0-0-3
Degree $2$
Conductor $2520$
Sign $0.605 + 0.795i$
Analytic cond. $1.25764$
Root an. cond. $1.12144$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−0.5 − 0.866i)10-s + (−1.5 − 0.866i)11-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)20-s − 1.73i·22-s + 25-s + (0.499 + 0.866i)28-s − 1.73i·29-s + (−1.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−0.5 − 0.866i)10-s + (−1.5 − 0.866i)11-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)20-s − 1.73i·22-s + 25-s + (0.499 + 0.866i)28-s − 1.73i·29-s + (−1.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6798826014\)
\(L(\frac12)\) \(\approx\) \(0.6798826014\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.485396425336285015466907215469, −8.034962519891171856258772190066, −7.55693494590602157134532958065, −6.84583540059739267997111764056, −5.75709249648064523663668108437, −5.11153683672593211588526782796, −4.19290316321355894485159732265, −3.63163838496177208330569650406, −2.56716251593342160146639648643, −0.37147482400292545584290763427, 1.63313637763301618604532342250, 2.65286807974832071738826059191, 3.39766792046319996410035203634, 4.52577536543458138214636770391, 5.07614372081784893850181640787, 5.71337868319051888157805895513, 7.03765255013557481153082138214, 7.69681086850212328557473798980, 8.670157763322775807472018485096, 9.090471495478327848395967257093

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