Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.34i·5-s + i·7-s + 0.896·11-s + 4·13-s + 6.31i·17-s i·19-s − 7.20·23-s − 6.19·25-s + 7.72i·29-s − 3.73i·31-s + 3.34·35-s − 11.7·37-s + 7.86i·41-s − 7.46i·43-s − 8.76·47-s + ⋯
L(s)  = 1  − 1.49i·5-s + 0.377i·7-s + 0.270·11-s + 1.10·13-s + 1.53i·17-s − 0.229i·19-s − 1.50·23-s − 1.23·25-s + 1.43i·29-s − 0.670i·31-s + 0.565·35-s − 1.92·37-s + 1.22i·41-s − 1.13i·43-s − 1.27·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $0.2455860758$
$L(\frac12)$  $\approx$  $0.2455860758$
$L(\frac{3}{2})$   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 \( 1 - iT \)
good5 \( 1 + 3.34iT - 5T^{2} \)
11 \( 1 - 0.896T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 6.31iT - 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + 7.20T + 23T^{2} \)
29 \( 1 - 7.72iT - 29T^{2} \)
31 \( 1 + 3.73iT - 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 - 7.86iT - 41T^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 + 8.76T + 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 - 3.86T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 7.46iT - 79T^{2} \)
83 \( 1 - 6.96T + 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 - 2.92T + 97T^{2} \)
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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

−8.370396884160719455392973117906, −8.000836490102786527319605522316, −6.81159081617470622220614554712, −6.04068041488584541056488887115, −5.55379041291329508762673420605, −4.71470761473313927994163615350, −4.00395566546333017923383030593, −3.32609406058859284291320912630, −1.78054439928037071553291603344, −1.41372709481705912490498029011, 0.05901414511772646209255841341, 1.56428453440115140464913586536, 2.55650877570963606628726976703, 3.36955645431725103839286091512, 3.90294705566882886451304348688, 4.88955284623074318543877979765, 5.97079457257095055792560618688, 6.36652001003433780985634569055, 7.14636559648961045370848629609, 7.59893116652608326405931443346

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