Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 $
Sign $0.850 - 0.525i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + (−1 + i)13-s + (1 + i)17-s + 25-s + 2i·29-s + (−1 − i)37-s + 2·41-s i·49-s + (1 − i)53-s + (−1 + i)65-s + (−1 + i)73-s + (1 + i)85-s − 2i·89-s + (−1 − i)97-s − 2i·109-s + ⋯
L(s)  = 1  + 5-s + (−1 + i)13-s + (1 + i)17-s + 25-s + 2i·29-s + (−1 − i)37-s + 2·41-s i·49-s + (1 − i)53-s + (−1 + i)65-s + (−1 + i)73-s + (1 + i)85-s − 2i·89-s + (−1 − i)97-s − 2i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.850 - 0.525i$
motivic weight  =  \(0\)
character  :  $\chi_{2880} (577, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2880,\ (\ :0),\ 0.850 - 0.525i)$
$L(\frac{1}{2})$  $\approx$  $1.453638547$
$L(\frac12)$  $\approx$  $1.453638547$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.054808555652830358511866688992, −8.451572014982720513571784719359, −7.28399161595257855237613723379, −6.87958905364385779778160339889, −5.82704323853452741440798708245, −5.33107753825434546417942632052, −4.37805761102425759304775548008, −3.36024568570201550238938503975, −2.28211909805043134554833880301, −1.46392526469955693743567413879, 1.00589148276618140205811909963, 2.43350896458390204821321676454, 2.93801762967853872878877683254, 4.27348818929452709643991685152, 5.21468584215457646832547000997, 5.70793900477979294766613176962, 6.53701648709589852388108456791, 7.53179085325822748549171207671, 7.947055717383514059880694882193, 9.122985483421329651393304358192

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