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  − 1.19i·5-s i·7-s − 1.62·11-s − 2.09·13-s + 0.585i·17-s − 5.89i·19-s + 7.09·23-s + 3.56·25-s − 7.08i·29-s + 6.72i·31-s − 1.19·35-s + 2.17·37-s − 0.164i·41-s + 5.75i·43-s − 3.41·47-s + ⋯
L(s)  = 1  − 0.536i·5-s − 0.377i·7-s − 0.491·11-s − 0.581·13-s + 0.142i·17-s − 1.35i·19-s + 1.47·23-s + 0.712·25-s − 1.31i·29-s + 1.20i·31-s − 0.202·35-s + 0.357·37-s − 0.0256i·41-s + 0.877i·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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$
$L(1)$  $\approx$  $1.225462840$
$L(\frac12)$  $\approx$  $1.225462840$
$L(\frac{3}{2})$   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 \( 1 + 1.19iT - 5T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
13 \( 1 + 2.09T + 13T^{2} \)
17 \( 1 - 0.585iT - 17T^{2} \)
19 \( 1 + 5.89iT - 19T^{2} \)
23 \( 1 - 7.09T + 23T^{2} \)
29 \( 1 + 7.08iT - 29T^{2} \)
31 \( 1 - 6.72iT - 31T^{2} \)
37 \( 1 - 2.17T + 37T^{2} \)
41 \( 1 + 0.164iT - 41T^{2} \)
43 \( 1 - 5.75iT - 43T^{2} \)
47 \( 1 + 3.41T + 47T^{2} \)
53 \( 1 + 3.29iT - 53T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 12.4iT - 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 3.97T + 73T^{2} \)
79 \( 1 + 0.702iT - 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 + 6.33iT - 89T^{2} \)
97 \( 1 + 6.58T + 97T^{2} \)
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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

−7.81826203052803057217035325384, −7.05340232445177798891165616744, −6.57956501100216842348765719143, −5.48391286862503603938263193861, −4.84595011740279269320612800731, −4.40764027588677753656834052436, −3.17651300696022795769662876076, −2.57153215485631563335810402541, −1.32092748651853921460799368334, −0.33316073626517574743890114605, 1.21302322791734309803042531012, 2.38564921050900223056319735961, 2.99719430008900515706932125422, 3.85431046611734685075030215825, 4.86768802581779777901662675470, 5.45974873474781357088098643323, 6.19548337867236469750684198542, 7.10432759545367246805751356143, 7.43598922637418240997050510535, 8.400413252965113134367708307334

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