Properties

Label 2-2880-40.29-c1-0-31
Degree $2$
Conductor $2880$
Sign $0.948 - 0.316i$
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  + (−1 + 2i)5-s + 2i·7-s − 2i·11-s + 2·13-s − 4i·17-s − 2i·19-s − 6i·23-s + (−3 − 4i)25-s + 4i·29-s + 8·31-s + (−4 − 2i)35-s + 10·37-s + 6·41-s + 4·43-s − 6i·47-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s + 0.755i·7-s − 0.603i·11-s + 0.554·13-s − 0.970i·17-s − 0.458i·19-s − 1.25i·23-s + (−0.600 − 0.800i)25-s + 0.742i·29-s + 1.43·31-s + (−0.676 − 0.338i)35-s + 1.64·37-s + 0.937·41-s + 0.609·43-s − 0.875i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.684269208\)
\(L(\frac12)\) \(\approx\) \(1.684269208\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−8.790429662327969257560757173359, −8.067914924638430663093665063319, −7.31654788048108036590523117773, −6.42462164491386394739469861875, −5.98514380004466445770661890363, −4.88028910859973911356463579475, −4.04225869926239842134364107290, −2.88949821771589606266114847200, −2.55061697328119929627044565661, −0.77875567907447025365915348453, 0.858207523796355079723462412511, 1.77787049368792626157617354394, 3.24544002672237736098148261049, 4.22179773891706083744489012477, 4.51964840539373667004252929861, 5.75346998660951441759653829124, 6.33234118097234750810990682118, 7.58155860841357234699511739885, 7.82463194663249580365793420036, 8.642625844249209508787148403382

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