Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.346 + 0.938i$
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

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.346 + 0.938i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.346 + 0.938i)$
$L(1)$  $\approx$  $1.347484857$
$L(\frac12)$  $\approx$  $1.347484857$
$L(\frac{3}{2})$   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

−8.232153323270213602399824451408, −7.42365506656986178601500942116, −6.87480990532988639549663425582, −6.04939183647311939813269400566, −5.05645708846153740756109955906, −4.61137398143543779293046210165, −3.25409970488395035486957139530, −3.05233875526128355099073309012, −1.46922200587654134678064935254, −0.41740603356420236361077799977, 1.25213980044446213158122507865, 2.26294584857771602178936975017, 3.26891414713724701425839374804, 4.10703123582461759917581470580, 4.90768364211216023826481875287, 5.70756001808077761159261549558, 6.48153144695275392054250109011, 7.22114126096701288704834347644, 8.094999903553939886077928132422, 8.420738540258208089560710645429

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