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  + (1 − i)2-s + i·3-s i·4-s + (1 + i)6-s − 9-s + 12-s + 2i·13-s + 16-s + (−1 + i)18-s + i·23-s + 25-s + (2 + 2i)26-s i·27-s i·29-s + (−1 − i)31-s + (1 − i)32-s + ⋯
L(s)  = 1  + (1 − i)2-s + i·3-s i·4-s + (1 + i)6-s − 9-s + 12-s + 2i·13-s + 16-s + (−1 + i)18-s + i·23-s + 25-s + (2 + 2i)26-s i·27-s i·29-s + (−1 − i)31-s + (1 − i)32-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  :  $\chi_{2001} (1172, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2001,\ (\ :0),\ 0.981 - 0.189i)$
$L(\frac{1}{2})$  $\approx$  $1.933768823$
$L(\frac12)$  $\approx$  $1.933768823$
$L(1)$   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 \( 1 - iT \)
23 \( 1 - iT \)
29 \( 1 + iT \)
good2 \( 1 + (-1 + i)T - iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - iT^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
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

−9.448714402438412153078856361613, −9.013583199878180017521722825469, −7.909930291327359386067163393486, −6.82301674379719791647076645185, −5.77487939191565230315492397537, −5.06090099359527545749738494172, −4.14873997327109448197734214665, −3.85620080643315639556510783474, −2.71534346064884293610064227614, −1.80863970995744292176853033784, 1.11946198236062556720060919945, 2.76570478491314705304799496869, 3.48871735581238695571082916185, 4.84606035679535512913514622259, 5.46915395071510417297680507734, 6.08930402049363758613191143349, 7.01173845027374617510831424490, 7.38519636203980971489244978722, 8.311054210908338613961222226780, 8.818386907003194553262343213001

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