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  + 1.93i·5-s i·7-s + 1.13·11-s − 3.19·13-s − 5.93i·17-s − 1.33i·19-s − 2.96·23-s + 1.26·25-s − 1.65i·29-s + 10.3i·31-s + 1.93·35-s + 0.538·37-s + 9.51i·41-s − 6.65i·43-s − 11.1·47-s + ⋯
L(s)  = 1  + 0.863i·5-s − 0.377i·7-s + 0.342·11-s − 0.885·13-s − 1.43i·17-s − 0.307i·19-s − 0.618·23-s + 0.253·25-s − 0.307i·29-s + 1.86i·31-s + 0.326·35-s + 0.0884·37-s + 1.48i·41-s − 1.01i·43-s − 1.62·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $0.8477029638$
$L(\frac12)$  $\approx$  $0.8477029638$
$L(\frac{3}{2})$   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 \( 1 - 1.93iT - 5T^{2} \)
11 \( 1 - 1.13T + 11T^{2} \)
13 \( 1 + 3.19T + 13T^{2} \)
17 \( 1 + 5.93iT - 17T^{2} \)
19 \( 1 + 1.33iT - 19T^{2} \)
23 \( 1 + 2.96T + 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 0.538T + 37T^{2} \)
41 \( 1 - 9.51iT - 41T^{2} \)
43 \( 1 + 6.65iT - 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 1.68iT - 53T^{2} \)
59 \( 1 - 1.82T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 9.53iT - 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 0.319T + 73T^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + 3.24T + 83T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 2.33T + 97T^{2} \)
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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

−8.256502274742312880780822581659, −7.46861934237490080612335208333, −6.84301141552312589081121494788, −6.56980135915698850939720775641, −5.33482909048169141434671519192, −4.82610505832826569727319168592, −3.87484983298682769698770803011, −2.99469685461390323352681921334, −2.43646209553396556481426927927, −1.14196646124642502667827024022, 0.22187799429114881856570736105, 1.54441485892614483064867657993, 2.25691247649721364447241371658, 3.43497846045044207092216055514, 4.23830060680147877428734788353, 4.86799628869235625631142932532, 5.73564054099630975269334886980, 6.21316821316653936844160228914, 7.16909106900430738768937428510, 7.972237962228745067736578536818

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