Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 5 $
Sign $0.367 - 0.930i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.86 − 1.23i)5-s + 0.746i·7-s + 5.36i·11-s − 2.92·13-s + 2.13i·17-s + 1.73i·19-s + 7.49i·23-s + (1.92 − 4.61i)25-s + 6.74i·29-s − 2.64·31-s + (0.925 + 1.38i)35-s − 1.07·37-s + 11.2·41-s − 7.44·43-s − 1.73i·47-s + ⋯
L(s)  = 1  + (0.832 − 0.554i)5-s + 0.282i·7-s + 1.61i·11-s − 0.811·13-s + 0.517i·17-s + 0.397i·19-s + 1.56i·23-s + (0.385 − 0.922i)25-s + 1.25i·29-s − 0.475·31-s + (0.156 + 0.234i)35-s − 0.176·37-s + 1.76·41-s − 1.13·43-s − 0.252i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.367 - 0.930i$
motivic weight  =  \(1\)
character  :  $\chi_{1440} (1009, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1440,\ (\ :1/2),\ 0.367 - 0.930i)\)
\(L(1)\)  \(\approx\)  \(1.653025560\)
\(L(\frac12)\)  \(\approx\)  \(1.653025560\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.86 + 1.23i)T \)
good7 \( 1 - 0.746iT - 7T^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 + 2.92T + 13T^{2} \)
17 \( 1 - 2.13iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 7.49iT - 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 7.44T + 43T^{2} \)
47 \( 1 + 1.73iT - 47T^{2} \)
53 \( 1 + 7.72T + 53T^{2} \)
59 \( 1 + 6.85iT - 59T^{2} \)
61 \( 1 + 6.45iT - 61T^{2} \)
67 \( 1 - 7.44T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 0.690iT - 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 + 5.85T + 83T^{2} \)
89 \( 1 - 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−9.580938275058193985757239193380, −9.174707535407614244715053969809, −8.036350565127538139882277348067, −7.27805225283611125989126819398, −6.41435981105441820893557054241, −5.33936244147344951200965525702, −4.90084037144086778086310483785, −3.72292975705491023808755920755, −2.29565540139803895615350775928, −1.55838631837197677616013559892, 0.65635533558555377851112369739, 2.34532674122059205921360451456, 3.04590940349524161916185500591, 4.27779972281684461919115233420, 5.36231635820950721486461726058, 6.12113906088393951399502042289, 6.82672722508742532412657688726, 7.74286462612622743521469499372, 8.660933133060370585081882790909, 9.423228750785570233719715049265

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