Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
Sign $0.987 + 0.160i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 16-s + (−1 − i)17-s + (1 − i)19-s + (1 + i)23-s + 32-s + (−1 − i)34-s + (1 − i)38-s + (1 + i)46-s + (−1 − i)47-s + i·49-s + 2i·53-s + (1 + i)61-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 16-s + (−1 − i)17-s + (1 − i)19-s + (1 + i)23-s + 32-s + (−1 − i)34-s + (1 − i)38-s + (1 + i)46-s + (−1 − i)47-s + i·49-s + 2i·53-s + (1 + i)61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.987 + 0.160i$
motivic weight  =  \(0\)
character  :  $\chi_{3600} (2557, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3600,\ (\ :0),\ 0.987 + 0.160i)$
$L(\frac{1}{2})$  $\approx$  $2.559874638$
$L(\frac12)$  $\approx$  $2.559874638$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−8.790992238917499576246919786578, −7.57513745641619228211733496290, −7.18856308503068707964419483434, −6.47241231574239751317512153207, −5.50470912544726460002159708056, −4.94797969430622192602973608200, −4.20836516324813963644486868443, −3.13808068165915260415013603914, −2.58104332158666329411768478999, −1.28524364473144289524865885834, 1.44904393535381829465750433600, 2.43928531236859538920031795290, 3.40537873563978453697267780075, 4.11156005976907006916252028381, 4.97765505645303978347063780143, 5.63142473548678773609038152371, 6.55904275422440354084720007923, 6.93020166853298988510859527003, 8.026537898858517693773409798944, 8.478989919133056169283558066796

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