Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.389 + 0.920i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.61·5-s i·7-s − 2.00i·11-s + 1.59i·13-s − 7.79i·17-s + 5.05·19-s + 7.14·23-s + 8.03·25-s − 4.94·29-s + 2.53i·31-s + 3.61i·35-s + 4.76i·37-s + 0.836i·41-s − 0.151·43-s + 0.795·47-s + ⋯
L(s)  = 1  − 1.61·5-s − 0.377i·7-s − 0.605i·11-s + 0.441i·13-s − 1.88i·17-s + 1.15·19-s + 1.49·23-s + 1.60·25-s − 0.917·29-s + 0.454i·31-s + 0.610i·35-s + 0.783i·37-s + 0.130i·41-s − 0.0230·43-s + 0.116·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.389 + 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6048,\ (\ :1/2),\ -0.389 + 0.920i)\)
\(L(1)\)  \(\approx\)  \(0.9750396921\)
\(L(\frac12)\)  \(\approx\)  \(0.9750396921\)
\(L(\frac{3}{2})\)   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 \( 1 + 3.61T + 5T^{2} \)
11 \( 1 + 2.00iT - 11T^{2} \)
13 \( 1 - 1.59iT - 13T^{2} \)
17 \( 1 + 7.79iT - 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 - 7.14T + 23T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 - 4.76iT - 37T^{2} \)
41 \( 1 - 0.836iT - 41T^{2} \)
43 \( 1 + 0.151T + 43T^{2} \)
47 \( 1 - 0.795T + 47T^{2} \)
53 \( 1 + 1.05T + 53T^{2} \)
59 \( 1 - 8.34iT - 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 + 0.655T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 + 3.81iT - 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 - 0.0444iT - 89T^{2} \)
97 \( 1 - 9.86T + 97T^{2} \)
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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

−7.57051737463386600566335976014, −7.34097995491689042224535102436, −6.76488602893485425784799946512, −5.54901554220662874709908118194, −4.83826783087186299190837427320, −4.22781214944915253306352865369, −3.22798647708880220757800962403, −2.95441710074077505141355492716, −1.19624810264879094942902615827, −0.33990013180427240336221088112, 0.965693270918538666975550817187, 2.17367899994139572118841539590, 3.39859778441756037106960193156, 3.69877043486675597399864149420, 4.64853543454247276291743612540, 5.32503375864933167090819983386, 6.20459139235140036513102623911, 7.12020302515626969926870088389, 7.62130704686069582757503813003, 8.171857829254914358166216678329

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