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  − 2.21i·5-s i·7-s − 0.614·11-s − 5.56·13-s + 0.585i·17-s + 3.89i·19-s − 0.850·23-s + 0.0963·25-s + 4.74i·29-s − 3.07i·31-s − 2.21·35-s + 2.17·37-s − 6.07i·41-s + 9.21i·43-s − 3.41·47-s + ⋯
L(s)  = 1  − 0.990i·5-s − 0.377i·7-s − 0.185·11-s − 1.54·13-s + 0.142i·17-s + 0.894i·19-s − 0.177·23-s + 0.0192·25-s + 0.880i·29-s − 0.551i·31-s − 0.374·35-s + 0.357·37-s − 0.949i·41-s + 1.40i·43-s − 0.498·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.132913368$
$L(\frac12)$  $\approx$  $1.132913368$
$L(\frac{3}{2})$   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 \( 1 + 2.21iT - 5T^{2} \)
11 \( 1 + 0.614T + 11T^{2} \)
13 \( 1 + 5.56T + 13T^{2} \)
17 \( 1 - 0.585iT - 17T^{2} \)
19 \( 1 - 3.89iT - 19T^{2} \)
23 \( 1 + 0.850T + 23T^{2} \)
29 \( 1 - 4.74iT - 29T^{2} \)
31 \( 1 + 3.07iT - 31T^{2} \)
37 \( 1 - 2.17T + 37T^{2} \)
41 \( 1 + 6.07iT - 41T^{2} \)
43 \( 1 - 9.21iT - 43T^{2} \)
47 \( 1 + 3.41T + 47T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 - 13.9T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 - 4.83iT - 67T^{2} \)
71 \( 1 + 5.33T + 71T^{2} \)
73 \( 1 + 9.28T + 73T^{2} \)
79 \( 1 + 1.29iT - 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 - 7.27T + 97T^{2} \)
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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

−8.037681291870861842998049058148, −7.60449499152718438598184608489, −6.83688173657454595450769553474, −5.92554432001766297641700165650, −5.16794728108004199204999415425, −4.63941010776604061777538875572, −3.88854853685782810234923855870, −2.85499499830057701871566955108, −1.88791899265750238759320607617, −0.874405994031680972316733739956, 0.34256551091459152926923513888, 2.01869548569584547670410123845, 2.67445716043996174366511224728, 3.29995797474232562393786406912, 4.46971878115254365059242741488, 5.05843716992672568202909606141, 5.90694296920907023652283000592, 6.72580832654775086262332114759, 7.19902256249125588929249701667, 7.83589451834374899195925830247

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