Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.769 + 0.639i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.646·5-s + i·7-s − 3.09i·11-s − 0.913i·13-s − 6.77i·17-s + 5.29·19-s − 2.53·23-s − 4.58·25-s − 9.93·29-s − 9.16i·31-s + 0.646i·35-s + 7.84i·37-s + 9.01i·41-s − 12.2·43-s + 5.65·47-s + ⋯
L(s)  = 1  + 0.288·5-s + 0.377i·7-s − 0.933i·11-s − 0.253i·13-s − 1.64i·17-s + 1.21·19-s − 0.528·23-s − 0.916·25-s − 1.84·29-s − 1.64i·31-s + 0.109i·35-s + 1.28i·37-s + 1.40i·41-s − 1.86·43-s + 0.825·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.769 + 0.639i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.769 + 0.639i)$
$L(1)$  $\approx$  $0.8915334205$
$L(\frac12)$  $\approx$  $0.8915334205$
$L(\frac{3}{2})$   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 \( 1 - 0.646T + 5T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 + 0.913iT - 13T^{2} \)
17 \( 1 + 6.77iT - 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
29 \( 1 + 9.93T + 29T^{2} \)
31 \( 1 + 9.16iT - 31T^{2} \)
37 \( 1 - 7.84iT - 37T^{2} \)
41 \( 1 - 9.01iT - 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 1.15T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 7.84T + 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 + 0.417T + 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + 15.9iT - 83T^{2} \)
89 \( 1 + 9.01iT - 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−8.035504258049253101907280762742, −7.57531078447517884682299084329, −6.65595810609423923837032819106, −5.71322756487132142364775883511, −5.44267034793239321672339330919, −4.38377751658160677626533512866, −3.33283641805455264744987512793, −2.70622735508539726818594871593, −1.54544819595916813189407937594, −0.24043438150772755665033707084, 1.52320737432118670071017131966, 2.10415685097924853108735394223, 3.57517809076743854991568371368, 3.96529367929462685086313269883, 5.11487223460716398355451331542, 5.69152882905237891345878663152, 6.58886655334502723494313006794, 7.32119160625194136875341226327, 7.86196699635700122206827002123, 8.782498143149273553243679451228

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