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Properties

Label 2-2520-2520.293-c0-0-5
Degree $2$
Conductor $2520$
Sign $-0.488 + 0.872i$
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  + (0.258 − 0.965i)2-s + (0.923 − 0.382i)3-s + (−0.866 − 0.499i)4-s + (−0.382 + 0.923i)5-s + (−0.130 − 0.991i)6-s + (−0.965 − 0.258i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.793 + 0.608i)10-s + (−0.991 − 0.130i)12-s + (1.78 − 0.478i)13-s + (−0.499 + 0.866i)14-s + i·15-s + (0.500 + 0.866i)16-s + (−0.500 − 0.866i)18-s − 1.98i·19-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.923 − 0.382i)3-s + (−0.866 − 0.499i)4-s + (−0.382 + 0.923i)5-s + (−0.130 − 0.991i)6-s + (−0.965 − 0.258i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.793 + 0.608i)10-s + (−0.991 − 0.130i)12-s + (1.78 − 0.478i)13-s + (−0.499 + 0.866i)14-s + i·15-s + (0.500 + 0.866i)16-s + (−0.500 − 0.866i)18-s − 1.98i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.488 + 0.872i$
Analytic conductor: \(1.25764\)
Root analytic conductor: \(1.12144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :0),\ -0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.483516485\)
\(L(\frac12)\) \(\approx\) \(1.483516485\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.78 + 0.478i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.98iT - T^{2} \)
23 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - 1.93iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T^{2} \)
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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

−8.801412793230008536966223584384, −8.408597590065243985560079225805, −7.25192474416397341040351909182, −6.59906797942888775331366974306, −5.86826541387495038180580006629, −4.39548085757972976122641934549, −3.65061578717162405634063148597, −3.06443986013305827898476193519, −2.39555793788565476279282198935, −0.894128212804250141113054157215, 1.54932880622514019554695582203, 3.32355636233335394373695929060, 3.76352274295314442825585887856, 4.44548380845832958495127961691, 5.63161703988737779302474916106, 6.11447590237670501239236203382, 7.17604299755248631269814574698, 8.029333527300343939543007698909, 8.497080945556804936003735471548, 9.095159063264960132718967871902

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