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  − 0.680i·5-s + i·7-s + 1.70·11-s + 5.59·13-s + 3.68i·17-s + 4.46i·19-s + 1.16·23-s + 4.53·25-s − 1.77i·29-s − 2.58i·31-s + 0.680·35-s + 1.36·37-s − 1.31i·41-s + 4.71i·43-s − 3.87·47-s + ⋯
L(s)  = 1  − 0.304i·5-s + 0.377i·7-s + 0.513·11-s + 1.55·13-s + 0.894i·17-s + 1.02i·19-s + 0.242·23-s + 0.907·25-s − 0.329i·29-s − 0.464i·31-s + 0.115·35-s + 0.223·37-s − 0.205i·41-s + 0.718i·43-s − 0.565·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $2.225452310$
$L(\frac12)$  $\approx$  $2.225452310$
$L(\frac{3}{2})$   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 \( 1 + 0.680iT - 5T^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 - 3.68iT - 17T^{2} \)
19 \( 1 - 4.46iT - 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 + 1.77iT - 29T^{2} \)
31 \( 1 + 2.58iT - 31T^{2} \)
37 \( 1 - 1.36T + 37T^{2} \)
41 \( 1 + 1.31iT - 41T^{2} \)
43 \( 1 - 4.71iT - 43T^{2} \)
47 \( 1 + 3.87T + 47T^{2} \)
53 \( 1 - 6.39iT - 53T^{2} \)
59 \( 1 + 0.948T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 + 1.28iT - 67T^{2} \)
71 \( 1 + 3.09T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 - 1.72iT - 79T^{2} \)
83 \( 1 + 7.69T + 83T^{2} \)
89 \( 1 - 0.962iT - 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.259773201547908323272049711792, −7.60438217599048707148162212108, −6.48187110429071266780368471255, −6.13338211073562008813197601535, −5.43069287439003193402140736950, −4.40697448603256885692742683975, −3.79762391284900707086499106029, −3.00109167013630908313798154049, −1.77138248341515693770124150247, −1.07161641328095084941525312788, 0.66957987455599207295428966358, 1.59338706102986386751699233289, 2.85981826188638407340317984516, 3.44146378390195373364976503198, 4.34099075674743132968982417236, 5.05536162465585668859907505990, 5.94069770580311479845689480304, 6.77270537693122628773492077507, 7.02833974420192668186279826618, 8.002742494157786552065155456864

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