Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.626 - 0.779i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.99i·5-s − 7-s + 2.13i·11-s + 0.665i·13-s − 7.04·17-s + 1.39i·19-s + 0.184·23-s − 3.94·25-s − 1.27i·29-s + 7.62·31-s + 2.99i·35-s − 6.69i·37-s + 0.274·41-s − 2.63i·43-s − 5.77·47-s + ⋯
L(s)  = 1  − 1.33i·5-s − 0.377·7-s + 0.645i·11-s + 0.184i·13-s − 1.70·17-s + 0.320i·19-s + 0.0384·23-s − 0.788·25-s − 0.236i·29-s + 1.36·31-s + 0.505i·35-s − 1.10i·37-s + 0.0428·41-s − 0.402i·43-s − 0.842·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.626 - 0.779i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.626 - 0.779i)$
$L(1)$  $\approx$  $1.076458196$
$L(\frac12)$  $\approx$  $1.076458196$
$L(\frac{3}{2})$   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 \( 1 + 2.99iT - 5T^{2} \)
11 \( 1 - 2.13iT - 11T^{2} \)
13 \( 1 - 0.665iT - 13T^{2} \)
17 \( 1 + 7.04T + 17T^{2} \)
19 \( 1 - 1.39iT - 19T^{2} \)
23 \( 1 - 0.184T + 23T^{2} \)
29 \( 1 + 1.27iT - 29T^{2} \)
31 \( 1 - 7.62T + 31T^{2} \)
37 \( 1 + 6.69iT - 37T^{2} \)
41 \( 1 - 0.274T + 41T^{2} \)
43 \( 1 + 2.63iT - 43T^{2} \)
47 \( 1 + 5.77T + 47T^{2} \)
53 \( 1 - 7.48iT - 53T^{2} \)
59 \( 1 - 9.03iT - 59T^{2} \)
61 \( 1 - 9.80iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 7.93T + 71T^{2} \)
73 \( 1 + 2.80T + 73T^{2} \)
79 \( 1 + 7.98T + 79T^{2} \)
83 \( 1 + 6.76iT - 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 0.107T + 97T^{2} \)
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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

−8.375822944062922134788917511136, −7.45695703894193856053412920067, −6.77768581567085229313155690569, −6.00647252457115133187145465321, −5.24607741144970297110282251183, −4.33822088886532796865711107708, −4.20794448922653639888849555763, −2.77748397368066467123686594195, −1.92932394663155921621564397087, −0.894234483912562051179571043720, 0.32235762301354009281713925685, 1.89529099931313700663338453438, 2.86569521845771541064653433314, 3.24550969050668296566540516173, 4.28350392871463377282347002819, 5.06866641586695547547907603608, 6.15771819766520238973512245869, 6.62283803375149706464890016894, 6.96491110036188830044117060166, 8.070812519712291687336774521234

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