Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2} $
Sign $0.816 - 0.577i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 1.41i·11-s + 13-s − 1.41i·17-s + 19-s + 1.41i·23-s + 1.41i·29-s + 31-s + 43-s + 1.41i·47-s − 1.41i·59-s + 61-s − 67-s − 1.41i·77-s − 1.41i·83-s + ⋯
L(s)  = 1  − 7-s + 1.41i·11-s + 13-s − 1.41i·17-s + 19-s + 1.41i·23-s + 1.41i·29-s + 31-s + 43-s + 1.41i·47-s − 1.41i·59-s + 61-s − 67-s − 1.41i·77-s − 1.41i·83-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.816 - 0.577i$
motivic weight  =  \(0\)
character  :  $\chi_{1800} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1800,\ (\ :0),\ 0.816 - 0.577i)$
$L(\frac{1}{2})$  $\approx$  $1.059826383$
$L(\frac12)$  $\approx$  $1.059826383$
$L(1)$   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 \( 1 \)
good7 \( 1 + T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.511175960404781618649421398952, −9.054463240882008076704081073943, −7.77444311268645063685427607118, −7.17833273073807590623599083198, −6.47279737514731729846813250396, −5.47907828137923836687614048949, −4.67081895796964342886463253640, −3.54627576406584498228687204239, −2.79957114848715855936262809137, −1.35574886675700468075506851066, 0.927879736999085279198485373471, 2.60133102228583573717590057644, 3.50745555893537077493993703579, 4.19746307343099414941982481620, 5.67741807178035927669926686561, 6.12224649955494156101339490121, 6.79555112695010867992420096465, 8.102257043226013423307724383503, 8.488175431717945062313005350818, 9.341590092977878629828543916595

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