Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.985 - 0.169i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·7-s − 4.24i·11-s − 6i·13-s − 4.24·23-s − 5·25-s − 4.24·29-s − 4i·31-s + 6i·37-s + 8.48i·41-s − 6·43-s − 8.48·47-s − 49-s + 12.7·53-s + 8.48i·59-s + 6i·61-s + ⋯
L(s)  = 1  − 0.377i·7-s − 1.27i·11-s − 1.66i·13-s − 0.884·23-s − 25-s − 0.787·29-s − 0.718i·31-s + 0.986i·37-s + 1.32i·41-s − 0.914·43-s − 1.23·47-s − 0.142·49-s + 1.74·53-s + 1.10i·59-s + 0.768i·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.985 - 0.169i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.985 - 0.169i)$
$L(1)$  $\approx$  $0.5046993528$
$L(\frac12)$  $\approx$  $0.5046993528$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 - 8.48iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 - 8.48iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.140630737945495921050259538070, −7.50768685595505047311050377660, −6.46384623162255478689977856472, −5.79252410295276734172617003258, −5.25287280722824407309816137669, −4.08020597178771150561004771456, −3.40449678285312499520422442216, −2.59418935690332892199898002178, −1.23445644086025279572211032659, −0.14228804874971602617736121744, 1.86315077615667525680878371579, 2.10110872134390969165841895213, 3.62116990776835524145782518484, 4.24225296431010947518190985975, 5.06279258406415346869821531979, 5.87472715297778390839506675012, 6.80142377286183488941458998056, 7.20020020880583669837973944067, 8.121038423024082749760834662193, 8.882573728797241316488891784238

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