Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.992 + 0.126i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.80i·5-s + 7-s − 3.51i·11-s + 4.87i·13-s + 3.16·17-s + 7.87i·19-s + 0.356·23-s − 2.87·25-s − 2.44i·29-s + 7·31-s − 2.80i·35-s − 5.87i·37-s + 10.1·41-s + 8.87i·43-s − 7.34·47-s + ⋯
L(s)  = 1  − 1.25i·5-s + 0.377·7-s − 1.06i·11-s + 1.35i·13-s + 0.766·17-s + 1.80i·19-s + 0.0743·23-s − 0.574·25-s − 0.454i·29-s + 1.25·31-s − 0.474i·35-s − 0.965i·37-s + 1.58·41-s + 1.35i·43-s − 1.07·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.992 + 0.126i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.992 + 0.126i)$
$L(1)$  $\approx$  $2.155674874$
$L(\frac12)$  $\approx$  $2.155674874$
$L(\frac{3}{2})$   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 \( 1 + 2.80iT - 5T^{2} \)
11 \( 1 + 3.51iT - 11T^{2} \)
13 \( 1 - 4.87iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 7.87iT - 19T^{2} \)
23 \( 1 - 0.356T + 23T^{2} \)
29 \( 1 + 2.44iT - 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + 5.87iT - 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 8.87iT - 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 12.9iT - 59T^{2} \)
61 \( 1 + 1.74iT - 61T^{2} \)
67 \( 1 - 14.6iT - 67T^{2} \)
71 \( 1 + 3.51T + 71T^{2} \)
73 \( 1 + 6.87T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 12.9iT - 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.060046634798059518693042764264, −7.66578610254732443470083138949, −6.44866636492399260327789056208, −5.87398484461037390998586654615, −5.23521424135846750295982520867, −4.31640117089909749497990616992, −3.90872543354598798261707271047, −2.72682214030002471775119679719, −1.54765269465188050733320215034, −0.942717416648375686788223624202, 0.69525190954627173304912226029, 2.05904705587709057306272508772, 2.89272636277178192599650976837, 3.37017194161124195379231474253, 4.64354753907491748302271178878, 5.07234229020346439490259192839, 6.10831633708939920613581670240, 6.74511099829727557114696387142, 7.41123957763301505782901332073, 7.85210346237590520220180772700

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