Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2} $
Sign $0.173 - 0.984i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s − 0.999·8-s + 9-s + (−1 + 1.73i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s − 2·17-s + (0.5 − 0.866i)18-s − 19-s + (0.999 + 1.73i)22-s + 0.999·24-s − 27-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s − 0.999·8-s + 9-s + (−1 + 1.73i)11-s + (0.499 + 0.866i)12-s + (−0.5 + 0.866i)16-s − 2·17-s + (0.5 − 0.866i)18-s − 19-s + (0.999 + 1.73i)22-s + 0.999·24-s − 27-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.173 - 0.984i$
motivic weight  =  \(0\)
character  :  $\chi_{1800} (1051, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1800,\ (\ :0),\ 0.173 - 0.984i)$
$L(\frac{1}{2})$  $\approx$  $0.2689208284$
$L(\frac12)$  $\approx$  $0.2689208284$
$L(1)$   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 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + 2T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.910023944342792120838150622733, −9.221672433493689494542748373485, −8.092650685060006443180412257156, −6.91336963670413770661341706546, −6.42028049841365022151502090050, −5.31283448836759103443258226680, −4.60099810867796520080950289180, −4.18826557465494761172551114441, −2.53060944909435527588667559895, −1.77606184994408508858255127027, 0.18068456797024635616045494218, 2.44656023537320876925877225840, 3.72573923948335484107591889285, 4.58211161765208124931401182762, 5.36970526932147481326199900939, 6.10830087856580903077164341736, 6.62827361643327361832839757443, 7.52731592307891002553681013779, 8.484988356238490620612424190935, 8.931788012563920906567232087794

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