Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.600 - 0.799i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.940i·5-s − 7-s + 5.98i·11-s + 6.59i·13-s + 2.64·17-s − 5.83i·19-s + 2.88·23-s + 4.11·25-s + 3.09i·29-s − 3.52·31-s + 0.940i·35-s + 0.213i·37-s + 1.63·41-s + 7.16i·43-s − 9.32·47-s + ⋯
L(s)  = 1  − 0.420i·5-s − 0.377·7-s + 1.80i·11-s + 1.82i·13-s + 0.640·17-s − 1.33i·19-s + 0.601·23-s + 0.823·25-s + 0.575i·29-s − 0.632·31-s + 0.158i·35-s + 0.0351i·37-s + 0.255·41-s + 1.09i·43-s − 1.36·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.600 - 0.799i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6048,\ (\ :1/2),\ -0.600 - 0.799i)\)
\(L(1)\)  \(\approx\)  \(1.214839079\)
\(L(\frac12)\)  \(\approx\)  \(1.214839079\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.940iT - 5T^{2} \)
11 \( 1 - 5.98iT - 11T^{2} \)
13 \( 1 - 6.59iT - 13T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
19 \( 1 + 5.83iT - 19T^{2} \)
23 \( 1 - 2.88T + 23T^{2} \)
29 \( 1 - 3.09iT - 29T^{2} \)
31 \( 1 + 3.52T + 31T^{2} \)
37 \( 1 - 0.213iT - 37T^{2} \)
41 \( 1 - 1.63T + 41T^{2} \)
43 \( 1 - 7.16iT - 43T^{2} \)
47 \( 1 + 9.32T + 47T^{2} \)
53 \( 1 + 7.51iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 + 1.48iT - 61T^{2} \)
67 \( 1 - 13.0iT - 67T^{2} \)
71 \( 1 + 1.54T + 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 - 8.74iT - 83T^{2} \)
89 \( 1 + 7.50T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−8.452826223651538342996908599749, −7.35807806449751873751689402205, −6.94052691910487303586633244763, −6.47719061386821230443415360359, −5.19528457235225505441215537254, −4.73512396667586381439972239547, −4.12889827641701406634066606293, −3.04356533318485907592833180354, −2.08886275384095098895541406835, −1.29120173278724428472512133604, 0.32676874169468905322602998586, 1.29671791650764008438797838401, 2.89118653480936011363180595510, 3.15480220115402865064214965875, 3.89823682150989750195452798968, 5.20350827626593313927213400767, 5.81509632248894515190770302770, 6.14516412590328490082566766370, 7.23370191222579187886287052707, 7.86428661715487366847813260970

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