Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.755 + 0.654i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 1.73i)7-s − 3.46i·11-s + 6.92i·17-s + 4·19-s + 3.46i·23-s + 5·25-s − 6·29-s + 4·31-s + 2·37-s + 6.92i·41-s − 3.46i·43-s + (1.00 + 6.92i)49-s − 6·53-s + 12·59-s − 13.8i·61-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)7-s − 1.04i·11-s + 1.68i·17-s + 0.917·19-s + 0.722i·23-s + 25-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 1.08i·41-s − 0.528i·43-s + (0.142 + 0.989i)49-s − 0.824·53-s + 1.56·59-s − 1.77i·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.755 + 0.654i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.755 + 0.654i)$
$L(1)$  $\approx$  $1.608895491$
$L(\frac12)$  $\approx$  $1.608895491$
$L(\frac{3}{2})$   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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.248255019414782111912352080542, −7.73354841966454197465585841455, −6.79381908308950977354078845526, −6.18747274383979713621610425701, −5.52796586846360439417760801881, −4.50334866010050679100304505693, −3.49587098587077795521985227194, −3.20728904815708636043613575844, −1.72722559933542220662194048639, −0.62770340494346190445203198598, 0.850528696633354334791286061642, 2.33926009880233913779483369028, 2.86827251153802823377210104329, 3.92320028349439863829764838679, 4.92799333860470475965942300229, 5.41854346842618749203109540012, 6.43564900400331234984549772955, 7.09079297781628177176323454343, 7.60421089041191712934912239510, 8.729042830694378557166007395628

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