Properties

Degree 2
Conductor $ 3 \cdot 23 \cdot 29 $
Sign $0.981 + 0.189i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 0.366i)2-s + (−0.866 + 0.5i)3-s − 0.732i·4-s + (−0.5 − 0.133i)6-s + (0.633 − 0.633i)8-s + (0.499 − 0.866i)9-s + (0.366 + 0.633i)12-s + i·13-s − 0.267·16-s + (0.499 − 0.133i)18-s i·23-s + (−0.232 + 0.866i)24-s + 25-s + (−0.366 + 0.366i)26-s + 0.999i·27-s + ⋯
L(s)  = 1  + (0.366 + 0.366i)2-s + (−0.866 + 0.5i)3-s − 0.732i·4-s + (−0.5 − 0.133i)6-s + (0.633 − 0.633i)8-s + (0.499 − 0.866i)9-s + (0.366 + 0.633i)12-s + i·13-s − 0.267·16-s + (0.499 − 0.133i)18-s i·23-s + (−0.232 + 0.866i)24-s + 25-s + (−0.366 + 0.366i)26-s + 0.999i·27-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2001\)    =    \(3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $0.981 + 0.189i$
motivic weight  =  \(0\)
character  :  $\chi_{2001} (1931, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2001,\ (\ :0),\ 0.981 + 0.189i)$
$L(\frac{1}{2})$  $\approx$  $1.084475138$
$L(\frac12)$  $\approx$  $1.084475138$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + iT^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - iT^{2} \)
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

−9.411785078926859909802694611019, −8.785602891370929991581599454193, −7.40134775538651311991797932163, −6.74931738114314809733842765572, −6.02092574292142631439586123884, −5.45284178150281076795511102209, −4.35922411813746182554232787230, −4.15565750339888851861985690707, −2.38730044178192699650682811334, −0.915586976546599552619892365333, 1.29856665633053463670349467816, 2.61322163691978724226645359645, 3.46677617296152623034344502172, 4.63429578908161776856387020092, 5.22513946429756488004515141268, 6.16787685037953061378513476204, 7.07821217227158131102089797499, 7.73204663749347350574215234803, 8.364215319625513677218455031018, 9.409876902396886075578402610684

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