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  + 1.93i·5-s i·7-s − 3.58·11-s + 5.92·13-s + 6.96i·17-s + 5.33i·19-s − 2.96·23-s + 1.26·25-s + 3.06i·29-s + 1.27i·31-s + 1.93·35-s + 9.65·37-s − 8.10i·41-s + 2.46i·43-s − 6.45·47-s + ⋯
L(s)  = 1  + 0.863i·5-s − 0.377i·7-s − 1.08·11-s + 1.64·13-s + 1.68i·17-s + 1.22i·19-s − 0.618·23-s + 0.253·25-s + 0.569i·29-s + 0.228i·31-s + 0.326·35-s + 1.58·37-s − 1.26i·41-s + 0.375i·43-s − 0.941·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.361848853$
$L(\frac12)$  $\approx$  $1.361848853$
$L(\frac{3}{2})$   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 \( 1 - 1.93iT - 5T^{2} \)
11 \( 1 + 3.58T + 11T^{2} \)
13 \( 1 - 5.92T + 13T^{2} \)
17 \( 1 - 6.96iT - 17T^{2} \)
19 \( 1 - 5.33iT - 19T^{2} \)
23 \( 1 + 2.96T + 23T^{2} \)
29 \( 1 - 3.06iT - 29T^{2} \)
31 \( 1 - 1.27iT - 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + 8.10iT - 41T^{2} \)
43 \( 1 - 2.46iT - 43T^{2} \)
47 \( 1 + 6.45T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + 6.34T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 + 6.26iT - 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 - 8.71iT - 79T^{2} \)
83 \( 1 + 7.96T + 83T^{2} \)
89 \( 1 - 9.04iT - 89T^{2} \)
97 \( 1 + 4.33T + 97T^{2} \)
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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

−8.137642631571338413830685365365, −7.83441116956573254896172881395, −6.84388822764113622567687075784, −6.10632496998768174929538469577, −5.81327067346947768112851415532, −4.64496699551388641936262505747, −3.67013589589753772891413350354, −3.37152793510649895902444211951, −2.17137013844230969225548366816, −1.29021626087301612774485678751, 0.36430875051775939586685352683, 1.31094906334666250221423650975, 2.58550167368898605666903843000, 3.12896348996832925208357201810, 4.49344539033291363606646753540, 4.74289110609783343389073807103, 5.72699839924404140538497171094, 6.18966099046153864401364786175, 7.22203527596218426857924025139, 7.909628034555338269874771757407

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