Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.779 + 0.626i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.50i·5-s + 7-s − 3.01i·11-s − 3.90i·13-s + 1.38·17-s + 4.79i·19-s + 5.06·23-s − 7.25·25-s − 4.91i·29-s − 1.13·31-s − 3.50i·35-s − 9.45i·37-s + 4.11·41-s − 1.51i·43-s + 10.7·47-s + ⋯
L(s)  = 1  − 1.56i·5-s + 0.377·7-s − 0.908i·11-s − 1.08i·13-s + 0.336·17-s + 1.09i·19-s + 1.05·23-s − 1.45·25-s − 0.911i·29-s − 0.204·31-s − 0.591i·35-s − 1.55i·37-s + 0.642·41-s − 0.231i·43-s + 1.57·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.779 + 0.626i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.779 + 0.626i)$
$L(1)$  $\approx$  $1.939471868$
$L(\frac12)$  $\approx$  $1.939471868$
$L(\frac{3}{2})$   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 \( 1 - T \)
good5 \( 1 + 3.50iT - 5T^{2} \)
11 \( 1 + 3.01iT - 11T^{2} \)
13 \( 1 + 3.90iT - 13T^{2} \)
17 \( 1 - 1.38T + 17T^{2} \)
19 \( 1 - 4.79iT - 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 + 4.91iT - 29T^{2} \)
31 \( 1 + 1.13T + 31T^{2} \)
37 \( 1 + 9.45iT - 37T^{2} \)
41 \( 1 - 4.11T + 41T^{2} \)
43 \( 1 + 1.51iT - 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 0.431iT - 53T^{2} \)
59 \( 1 - 7.40iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 - 3.36iT - 67T^{2} \)
71 \( 1 - 6.26T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + 0.818T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−7.906160650127367072163979497998, −7.38064468718697120884040923020, −6.01626066622423739173224513025, −5.62167550200047941206602455885, −5.04562023087597780335091326478, −4.15551813386716660101334561895, −3.46938465717785764299637352667, −2.33166267519215332653749031767, −1.17701477855712497661007101952, −0.55030907290514497197411751403, 1.35416911155772787478846040610, 2.41667301605796098525325764047, 2.94534034144632051361694737611, 3.95285461611732286830926113178, 4.69050354640953909102381905893, 5.48582320777036646486076633230, 6.57200902529372915718819516506, 6.94321017238052101729412143438, 7.34120527676348929637540421458, 8.266501453236595982268475350327

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