Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.965 - 0.258i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73i·5-s − 7-s − 5.46i·11-s − 6.73i·13-s − 2·17-s − 1.26i·19-s − 3.46·23-s − 2.46·25-s − 1.46i·29-s − 4·31-s + 2.73i·35-s − 1.46i·37-s + 2·41-s + 5.46i·43-s + 2.92·47-s + ⋯
L(s)  = 1  − 1.22i·5-s − 0.377·7-s − 1.64i·11-s − 1.86i·13-s − 0.485·17-s − 0.290i·19-s − 0.722·23-s − 0.492·25-s − 0.271i·29-s − 0.718·31-s + 0.461i·35-s − 0.240i·37-s + 0.312·41-s + 0.833i·43-s + 0.427·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.965 - 0.258i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2017, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.965 - 0.258i)$
$L(1)$  $\approx$  $1.091175556$
$L(\frac12)$  $\approx$  $1.091175556$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 2.73iT - 5T^{2} \)
11 \( 1 + 5.46iT - 11T^{2} \)
13 \( 1 + 6.73iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 1.26iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 1.46iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 1.46iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 5.46iT - 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 9.66iT - 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 2.92T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 5.66iT - 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−8.294223623181377900101168035785, −7.51146070424681157701999803687, −6.39138192700564970104563657955, −5.62037114378331800433256324955, −5.30950037888657056992897584542, −4.19023035457148739337713994622, −3.39418598480175409499983283713, −2.54175496718756926735722025659, −1.03903121379069734055922913777, −0.34492076556418182734895864822, 1.87035801753806654882220904304, 2.29918942516118968933742609884, 3.52535459676349387984507876790, 4.20620152303346507327336569959, 4.98155777499815363662452736756, 6.22346244067086025248847766358, 6.71746048674274947607776783369, 7.17850952055671448721370334649, 7.892389565834149131938256886691, 9.113420112540729419049667009918

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