Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.951 - 0.309i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.29i·5-s + 7-s + 1.60i·11-s + 6.02i·13-s − 3.16·17-s − 3.07i·19-s + 5.95·23-s − 13.4·25-s + 3.53i·29-s + 2.08·31-s − 4.29i·35-s + 4.48i·37-s + 6.69·41-s + 12.2i·43-s + 3.82·47-s + ⋯
L(s)  = 1  − 1.92i·5-s + 0.377·7-s + 0.484i·11-s + 1.67i·13-s − 0.766·17-s − 0.706i·19-s + 1.24·23-s − 2.69·25-s + 0.657i·29-s + 0.375·31-s − 0.726i·35-s + 0.736i·37-s + 1.04·41-s + 1.87i·43-s + 0.558·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.951 - 0.309i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.951 - 0.309i)$
$L(1)$  $\approx$  $1.747756961$
$L(\frac12)$  $\approx$  $1.747756961$
$L(\frac{3}{2})$   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 \( 1 + 4.29iT - 5T^{2} \)
11 \( 1 - 1.60iT - 11T^{2} \)
13 \( 1 - 6.02iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 3.07iT - 19T^{2} \)
23 \( 1 - 5.95T + 23T^{2} \)
29 \( 1 - 3.53iT - 29T^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 - 4.48iT - 37T^{2} \)
41 \( 1 - 6.69T + 41T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 - 3.82T + 47T^{2} \)
53 \( 1 - 8.63iT - 53T^{2} \)
59 \( 1 + 1.55iT - 59T^{2} \)
61 \( 1 - 3.64iT - 61T^{2} \)
67 \( 1 + 0.772iT - 67T^{2} \)
71 \( 1 - 7.36T + 71T^{2} \)
73 \( 1 - 1.32T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 7.08iT - 83T^{2} \)
89 \( 1 + 9.73T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.341694812441024951474562019470, −7.41397389186502773834208036920, −6.76288907479535268602705572480, −5.88147527398660277793833236818, −4.92434651315496129158260694135, −4.58602908438366528167497845082, −4.12492958717606087367847774231, −2.65177789434095954135858929163, −1.64638275559828341836632454807, −1.00693515594459741845397051248, 0.50561342925499237006933202823, 2.08097445481705132559141053669, 2.81888120562780622085077660121, 3.40771541561357110877065369316, 4.19928357853064438021837591334, 5.48848725694052251013446437042, 5.84346088463692107062241567527, 6.75657798520392570730256764894, 7.25818124249134534791853655663, 7.939018605103857466007195887694

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