Properties

Label 2-2880-8.5-c1-0-24
Degree $2$
Conductor $2880$
Sign $0.707 + 0.707i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
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  + i·5-s − 6i·13-s + 2i·19-s + 6·23-s − 25-s + 6i·29-s − 6i·37-s − 6·41-s − 8i·43-s + 6·47-s − 7·49-s − 6i·53-s − 12i·59-s − 12i·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.66i·13-s + 0.458i·19-s + 1.25·23-s − 0.200·25-s + 1.11i·29-s − 0.986i·37-s − 0.937·41-s − 1.21i·43-s + 0.875·47-s − 49-s − 0.824i·53-s − 1.56i·59-s − 1.53i·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.651159801\)
\(L(\frac12)\) \(\approx\) \(1.651159801\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.573091330886350399659647027354, −7.945330926207440916273343694046, −7.16258733738589617729740237835, −6.48922672055492464537477326866, −5.45350893226106825946899862817, −5.03821679825575861947731981153, −3.62777866060166136172783597331, −3.15150350402670343791280774122, −2.00814878055854722772814245621, −0.60935198174608818132866900175, 1.07744917579644821256237177127, 2.15290710932403608688646194148, 3.21835517779007736385053037554, 4.38637631019046432340169399509, 4.75029737616232500864283352058, 5.86713601413244357990765643181, 6.63781115949250136543882480737, 7.28739593104935365220407784665, 8.201738405221438610130321612715, 8.995023667265877932221464534425

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