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  − 0.896i·5-s i·7-s − 3.34·11-s + 4·13-s − 3.48i·17-s + i·19-s − 0.138·23-s + 4.19·25-s − 2.07i·29-s + 0.267i·31-s − 0.896·35-s − 8.26·37-s − 9.28i·41-s + 0.535i·43-s − 5.93·47-s + ⋯
L(s)  = 1  − 0.400i·5-s − 0.377i·7-s − 1.00·11-s + 1.10·13-s − 0.845i·17-s + 0.229i·19-s − 0.0289·23-s + 0.839·25-s − 0.384i·29-s + 0.0481i·31-s − 0.151·35-s − 1.35·37-s − 1.44i·41-s + 0.0817i·43-s − 0.865·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 + 0.707i)$
$L(1)$  $\approx$  $1.167965689$
$L(\frac12)$  $\approx$  $1.167965689$
$L(\frac{3}{2})$   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 \( 1 + 0.896iT - 5T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 3.48iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + 0.138T + 23T^{2} \)
29 \( 1 + 2.07iT - 29T^{2} \)
31 \( 1 - 0.267iT - 31T^{2} \)
37 \( 1 + 8.26T + 37T^{2} \)
41 \( 1 + 9.28iT - 41T^{2} \)
43 \( 1 - 0.535iT - 43T^{2} \)
47 \( 1 + 5.93T + 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 - 1.03T + 59T^{2} \)
61 \( 1 - 8.39T + 61T^{2} \)
67 \( 1 + 2.92iT - 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 0.535iT - 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.908154111795597726965787154950, −7.08444248696093038785889500295, −6.48723243559062464455379428916, −5.45204836721418466013882865172, −5.09549175582322548859210678425, −4.11145761865523032867797163975, −3.37687235630442993191766630102, −2.46372626430393017834904340794, −1.36855607950418212611061209078, −0.30943520918012587124233452503, 1.24025422444832665408212657608, 2.27228068855398728491646747521, 3.13926490914665287324036868599, 3.78460078281109435244200388738, 4.87930780102829960020257272760, 5.44624446160485420380927389320, 6.32888062657961872759495047141, 6.76980053249088329149143262000, 7.74663986854685002140827243848, 8.361144655753430438098598526114

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