Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.940 + 0.340i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.436·5-s i·7-s − 2.73i·11-s − 1.64i·13-s − 5.22i·17-s + 5.99·19-s − 7.68·23-s − 4.80·25-s + 4.94·29-s − 9.77i·31-s + 0.436i·35-s + 3.81i·37-s + 9.74i·41-s + 8.84·43-s + 4.54·47-s + ⋯
L(s)  = 1  − 0.195·5-s − 0.377i·7-s − 0.825i·11-s − 0.456i·13-s − 1.26i·17-s + 1.37·19-s − 1.60·23-s − 0.961·25-s + 0.918·29-s − 1.75i·31-s + 0.0737i·35-s + 0.627i·37-s + 1.52i·41-s + 1.34·43-s + 0.663·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.940 + 0.340i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6048,\ (\ :1/2),\ -0.940 + 0.340i)\)
\(L(1)\)  \(\approx\)  \(0.9275607365\)
\(L(\frac12)\)  \(\approx\)  \(0.9275607365\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.436T + 5T^{2} \)
11 \( 1 + 2.73iT - 11T^{2} \)
13 \( 1 + 1.64iT - 13T^{2} \)
17 \( 1 + 5.22iT - 17T^{2} \)
19 \( 1 - 5.99T + 19T^{2} \)
23 \( 1 + 7.68T + 23T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 + 9.77iT - 31T^{2} \)
37 \( 1 - 3.81iT - 37T^{2} \)
41 \( 1 - 9.74iT - 41T^{2} \)
43 \( 1 - 8.84T + 43T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 - 6.26iT - 61T^{2} \)
67 \( 1 + 9.42T + 67T^{2} \)
71 \( 1 + 9.76T + 71T^{2} \)
73 \( 1 + 9.14T + 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 - 0.227iT - 83T^{2} \)
89 \( 1 - 2.46iT - 89T^{2} \)
97 \( 1 + 3.37T + 97T^{2} \)
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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

−7.85941383150831786991551124578, −7.22793859862324064809027901531, −6.15218695812048383259208625991, −5.78832311873814182602782603807, −4.81040761556321421756262400998, −4.09276927842889652427664045543, −3.22267088962048433393426324208, −2.55243988768263743836440909189, −1.20410448213283119308899493518, −0.24583708405724929511078954041, 1.42467786521355378012279491194, 2.15984693792710391208844639719, 3.22627689412645147787981627331, 4.06705455962787712031698670251, 4.65887544215289003964525961030, 5.74067864716546393042804108014, 6.06151417599661252307220368759, 7.19894726333928426802509231219, 7.53104225480611220982910423675, 8.436754782235299455703730731906

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