Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.763 - 0.646i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.328 + 1.37i)2-s + (−1.78 + 0.903i)4-s + 0.512i·5-s − 7-s + (−1.82 − 2.15i)8-s + (−0.704 + 0.168i)10-s − 1.82i·11-s − 1.80i·13-s + (−0.328 − 1.37i)14-s + (2.36 − 3.22i)16-s + 8.11·17-s − 3.43i·19-s + (−0.462 − 0.914i)20-s + (2.51 − 0.600i)22-s − 3.65·23-s + ⋯
L(s)  = 1  + (0.232 + 0.972i)2-s + (−0.892 + 0.451i)4-s + 0.229i·5-s − 0.377·7-s + (−0.646 − 0.763i)8-s + (−0.222 + 0.0531i)10-s − 0.551i·11-s − 0.500i·13-s + (−0.0877 − 0.367i)14-s + (0.592 − 0.805i)16-s + 1.96·17-s − 0.789i·19-s + (−0.103 − 0.204i)20-s + (0.536 − 0.128i)22-s − 0.762·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.763 - 0.646i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.763 - 0.646i)$
$L(1)$  $\approx$  $1.585128524$
$L(\frac12)$  $\approx$  $1.585128524$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.328 - 1.37i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 0.512iT - 5T^{2} \)
11 \( 1 + 1.82iT - 11T^{2} \)
13 \( 1 + 1.80iT - 13T^{2} \)
17 \( 1 - 8.11T + 17T^{2} \)
19 \( 1 + 3.43iT - 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 7.98iT - 29T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 - 8.44iT - 37T^{2} \)
41 \( 1 + 2.30T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 8.87iT - 53T^{2} \)
59 \( 1 + 2.50iT - 59T^{2} \)
61 \( 1 - 5.31iT - 61T^{2} \)
67 \( 1 + 6.44iT - 67T^{2} \)
71 \( 1 - 9.10T + 71T^{2} \)
73 \( 1 - 9.24T + 73T^{2} \)
79 \( 1 - 3.64T + 79T^{2} \)
83 \( 1 + 4.53iT - 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−9.626277398638281055395839767942, −8.448028844569745291509171229299, −7.958166838273666566753683465398, −7.12235106359541059531685922917, −6.23177279003531010599862236563, −5.63700673628159355195280635446, −4.72436959861321330444494608672, −3.58505425836269565315796404151, −2.88509891759936813643539302712, −0.74454432723300926003519802207, 1.07183800269576097156063959561, 2.18876704322959046306628229030, 3.40698414807423521583539643021, 4.03570541502080047752551167212, 5.22549980261484405888328461660, 5.74671120993647358573458312141, 6.99897971821877338548665524584, 7.904773579136874401540206681297, 8.899689024552940849633244095111, 9.470883127960428465395749789233

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