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

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

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  =  $(2,\ 1800,\ (\ :0),\ 0.816 + 0.577i)$
$L(\frac{1}{2})$  $\approx$  $1.230051638$
$L(\frac12)$  $\approx$  $1.230051638$
$L(1)$   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 \( 1 \)
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

−9.481643109819209187920328009155, −8.516328304255330979798076045165, −7.77888556726321503794629606575, −7.25221067118758668698812393630, −6.05150573135710332496011069713, −5.27200937995422452314448589512, −4.62705114421934455860040065344, −3.37350855651480880371050693834, −2.50054492117105069112874598586, −1.04320301703188116681147743593, 1.56505875568933193688038560825, 2.43891429590937091663326239392, 3.80244267575712987763138962422, 4.83784542716465231509154655959, 5.15345747603836662233310407075, 6.55212368187699617938367334150, 7.15616098528516266801275248004, 8.058916373388292310969097090446, 8.571117989757483871185316019410, 9.726849853760162435945790524760

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