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  + 1.29i·5-s i·7-s − 2.21·11-s − 1.03·13-s + 2.24i·17-s + 8.63i·19-s + 8.96·23-s + 3.31·25-s − 3.26i·29-s + 4.37i·31-s + 1.29·35-s − 7.21·37-s + 7.58i·41-s − 12.9i·43-s − 5.24·47-s + ⋯
L(s)  = 1  + 0.579i·5-s − 0.377i·7-s − 0.668·11-s − 0.287·13-s + 0.544i·17-s + 1.98i·19-s + 1.87·23-s + 0.663·25-s − 0.606i·29-s + 0.786i·31-s + 0.219·35-s − 1.18·37-s + 1.18i·41-s − 1.97i·43-s − 0.764·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.075323350$
$L(\frac12)$  $\approx$  $1.075323350$
$L(\frac{3}{2})$   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 \( 1 - 1.29iT - 5T^{2} \)
11 \( 1 + 2.21T + 11T^{2} \)
13 \( 1 + 1.03T + 13T^{2} \)
17 \( 1 - 2.24iT - 17T^{2} \)
19 \( 1 - 8.63iT - 19T^{2} \)
23 \( 1 - 8.96T + 23T^{2} \)
29 \( 1 + 3.26iT - 29T^{2} \)
31 \( 1 - 4.37iT - 31T^{2} \)
37 \( 1 + 7.21T + 37T^{2} \)
41 \( 1 - 7.58iT - 41T^{2} \)
43 \( 1 + 12.9iT - 43T^{2} \)
47 \( 1 + 5.24T + 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 - 2.46T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 4.46iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 9.14iT - 79T^{2} \)
83 \( 1 - 9.96T + 83T^{2} \)
89 \( 1 + 3.48iT - 89T^{2} \)
97 \( 1 + 6.30T + 97T^{2} \)
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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

−8.243478282698189222126197181789, −7.63317125299786997944132903284, −6.89109913120403543389292965605, −6.37889710539491762717867929399, −5.37033772361515534607278577891, −4.88930608032903979115990502881, −3.71005801171819039938516385177, −3.25945959454023476091665569336, −2.23012232839742621915700347366, −1.22857444067921497098430266054, 0.28641693946923995909113954676, 1.35494403204187320746342895214, 2.71328894739271697873788421815, 2.98013145392206223436608203683, 4.41579510124662145172317335687, 5.01842171791802164630730296433, 5.36065702743579594952016082901, 6.52649010485192274071932572596, 7.11297181874972611219095261457, 7.74657865550751761078556638975

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