Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.752 - 0.658i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.111 + 0.111i)5-s − 7-s + (−3.61 − 3.61i)11-s + (−1.94 + 1.94i)13-s − 4.79i·17-s + (−3.03 − 3.03i)19-s + 6.58i·23-s + 4.97i·25-s + (1.53 + 1.53i)29-s + 3.26i·31-s + (0.111 − 0.111i)35-s + (1.05 + 1.05i)37-s − 1.26·41-s + (−0.484 + 0.484i)43-s + 11.2·47-s + ⋯
L(s)  = 1  + (−0.0498 + 0.0498i)5-s − 0.377·7-s + (−1.08 − 1.08i)11-s + (−0.539 + 0.539i)13-s − 1.16i·17-s + (−0.695 − 0.695i)19-s + 1.37i·23-s + 0.995i·25-s + (0.284 + 0.284i)29-s + 0.586i·31-s + (0.0188 − 0.0188i)35-s + (0.173 + 0.173i)37-s − 0.197·41-s + (−0.0738 + 0.0738i)43-s + 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.752 - 0.658i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.752 - 0.658i)$
$L(1)$  $\approx$  $1.164429225$
$L(\frac12)$  $\approx$  $1.164429225$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + (0.111 - 0.111i)T - 5iT^{2} \)
11 \( 1 + (3.61 + 3.61i)T + 11iT^{2} \)
13 \( 1 + (1.94 - 1.94i)T - 13iT^{2} \)
17 \( 1 + 4.79iT - 17T^{2} \)
19 \( 1 + (3.03 + 3.03i)T + 19iT^{2} \)
23 \( 1 - 6.58iT - 23T^{2} \)
29 \( 1 + (-1.53 - 1.53i)T + 29iT^{2} \)
31 \( 1 - 3.26iT - 31T^{2} \)
37 \( 1 + (-1.05 - 1.05i)T + 37iT^{2} \)
41 \( 1 + 1.26T + 41T^{2} \)
43 \( 1 + (0.484 - 0.484i)T - 43iT^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + (-4.00 + 4.00i)T - 53iT^{2} \)
59 \( 1 + (-7.61 - 7.61i)T + 59iT^{2} \)
61 \( 1 + (-5.44 + 5.44i)T - 61iT^{2} \)
67 \( 1 + (-0.897 - 0.897i)T + 67iT^{2} \)
71 \( 1 - 2.83iT - 71T^{2} \)
73 \( 1 - 15.7iT - 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (-7.57 + 7.57i)T - 83iT^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−8.717274218113482259197989199270, −7.60050497588910335325841596229, −7.21483032469506190984686311711, −6.35026481002958118189532708028, −5.39574370478599601209683172509, −5.00088307689120415193363100648, −3.81626014694029260908349218449, −3.01409401150566478112102850911, −2.27939431835683590757777603498, −0.78099058435477991619523745346, 0.45036326084336618708487639986, 2.12284730447345898829612703495, 2.60012438734118238370848527491, 3.90536579910806408255895707727, 4.50038822973148837019806431185, 5.39595372112594770601676306715, 6.17605524648171275200706433228, 6.86171710701079954246551395942, 7.80896123521981222340598132853, 8.178070455861407609943106946369

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