Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2} $
Sign $-0.232 - 0.972i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.232 - 0.972i$
motivic weight  =  \(0\)
character  :  $\chi_{1800} (499, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1800,\ (\ :0),\ -0.232 - 0.972i)$
$L(\frac{1}{2})$  $\approx$  $0.3983499006$
$L(\frac12)$  $\approx$  $0.3983499006$
$L(1)$   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 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + 2T + T^{2} \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.911528475078839110408799632546, −8.637867402844645434691265742481, −8.199883273305099593035734116704, −7.17673155976640400362408448939, −6.56875302196552206070886735752, −6.06203510735435564077399046923, −5.01416789179383666717981765888, −4.17447292961991938419969427431, −2.25859564692299210050947729146, −1.42860540125976050099974465698, 0.45395563423841967327361832375, 1.97058776441899405761209405845, 3.32332478407927401159022863107, 4.10187096955129550011389501728, 5.12881101680909443395623677804, 6.30625675643013660697211699987, 6.72438027726173487667877908285, 7.79556411831064873483009761563, 8.832899843128343455092852818108, 9.147237538689713614236337955875

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