Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.727 - 0.686i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.21·5-s + i·7-s + 2.42i·11-s − 1.11i·13-s − 0.701i·17-s − 0.938·19-s − 4.30·23-s − 0.102·25-s − 4.26·29-s + 7.02i·31-s + 2.21i·35-s + 3.12i·37-s − 0.157i·41-s − 7.08·43-s + 0.867·47-s + ⋯
L(s)  = 1  + 0.989·5-s + 0.377i·7-s + 0.730i·11-s − 0.309i·13-s − 0.170i·17-s − 0.215·19-s − 0.897·23-s − 0.0204·25-s − 0.791·29-s + 1.26i·31-s + 0.374i·35-s + 0.513i·37-s − 0.0246i·41-s − 1.08·43-s + 0.126·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.727 - 0.686i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.727 - 0.686i)$
$L(1)$  $\approx$  $1.152514397$
$L(\frac12)$  $\approx$  $1.152514397$
$L(\frac{3}{2})$   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 \( 1 - 2.21T + 5T^{2} \)
11 \( 1 - 2.42iT - 11T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 + 0.701iT - 17T^{2} \)
19 \( 1 + 0.938T + 19T^{2} \)
23 \( 1 + 4.30T + 23T^{2} \)
29 \( 1 + 4.26T + 29T^{2} \)
31 \( 1 - 7.02iT - 31T^{2} \)
37 \( 1 - 3.12iT - 37T^{2} \)
41 \( 1 + 0.157iT - 41T^{2} \)
43 \( 1 + 7.08T + 43T^{2} \)
47 \( 1 - 0.867T + 47T^{2} \)
53 \( 1 + 3.91T + 53T^{2} \)
59 \( 1 - 7.28iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 - 7.02T + 73T^{2} \)
79 \( 1 + 6.65iT - 79T^{2} \)
83 \( 1 - 8.41iT - 83T^{2} \)
89 \( 1 - 7.34iT - 89T^{2} \)
97 \( 1 - 7.99T + 97T^{2} \)
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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

−8.351502050128772051027558450906, −7.65186586828742041711426316035, −6.82258763823484395706113826895, −6.19686622252885222117897090747, −5.48855414784291594728759683501, −4.90578166015155402402183502333, −3.96900172404923459133670294158, −2.97604807867144771524684439824, −2.12723393040136964361158182728, −1.44163977282570425512412250832, 0.26011054084795520428102990914, 1.61621265876316809580060611577, 2.26226368734462327770193587899, 3.37350586102892735928081303449, 4.08765827413360421666438170466, 4.99204083241242242046264261603, 5.91280312419149495977808627288, 6.14048838907830694891002363838, 7.05810529644761388844770796878, 7.86138620748581655292298135350

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