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  + (−1.36 − 1.36i)2-s + (0.866 + 0.5i)3-s + 2.73i·4-s + (−0.499 − 1.86i)6-s + (2.36 − 2.36i)8-s + (0.499 + 0.866i)9-s + (−1.36 + 2.36i)12-s + i·13-s − 3.73·16-s + (0.5 − 1.86i)18-s i·23-s + (3.23 − 0.866i)24-s + 25-s + (1.36 − 1.36i)26-s + 0.999i·27-s + ⋯
L(s)  = 1  + (−1.36 − 1.36i)2-s + (0.866 + 0.5i)3-s + 2.73i·4-s + (−0.499 − 1.86i)6-s + (2.36 − 2.36i)8-s + (0.499 + 0.866i)9-s + (−1.36 + 2.36i)12-s + i·13-s − 3.73·16-s + (0.5 − 1.86i)18-s i·23-s + (3.23 − 0.866i)24-s + 25-s + (1.36 − 1.36i)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  =  $(2,\ 2001,\ (\ :0),\ 0.981 + 0.189i)$
$L(\frac{1}{2})$  $\approx$  $0.7869477382$
$L(\frac12)$  $\approx$  $0.7869477382$
$L(1)$   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 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
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 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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

−9.364731287060467008354394539108, −8.725705416815685779339773465196, −8.282348751421814410122542859343, −7.39661435503404240555190115090, −6.62678357677016197280100112272, −4.71053927755874912365606573989, −4.08342838137541537382530050618, −3.06314956301727082962698382275, −2.41534826216741107223804090846, −1.37099445475855887632262427722, 0.915726320461567854752598713572, 2.05922790361499140130650471231, 3.39657362086873662715205391487, 4.93061985556174955312500072481, 5.69867129524085286195481193168, 6.70254386874767697018534394739, 7.10412934105257900290598490143, 8.033831685052919770827941175704, 8.326749155444075907232842323589, 9.131549916224690491095738961234

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