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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 + 0.707i)$
$L(1)$  $\approx$  $2.028667621$
$L(\frac12)$  $\approx$  $2.028667621$
$L(\frac{3}{2})$   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

−8.063329238246267115351513967435, −7.29681922056566914484813764161, −6.51994234714341042483467150217, −5.90717042563767409064823284389, −5.10203024039185233904085498511, −4.29483073257915870848780039507, −3.61626361889285964598719170914, −2.71782396818447004851299855948, −1.59676175212266313027200696047, −0.66327007916709642747725611270, 0.910430080580916538077047288434, 2.00023693758714212507218213439, 2.97620546487330044795037745849, 3.61243715586064724818549675784, 4.55039638349690451803034052650, 5.40289550755453513517518778578, 6.04921055341478705961194598676, 6.65466125800075259619059265532, 7.53210609357689987492261295121, 8.171982718320611146971175415274

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