Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 5^{2} $
Sign $0.894 - 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·7-s − 4·11-s − 2i·13-s − 6i·17-s + 4·19-s + 2·29-s + 4·31-s + 2i·37-s − 2·41-s + 4i·43-s + 8i·47-s − 9·49-s − 10i·53-s − 4·59-s + 6·61-s + ⋯
L(s)  = 1  + 1.51i·7-s − 1.20·11-s − 0.554i·13-s − 1.45i·17-s + 0.917·19-s + 0.371·29-s + 0.718·31-s + 0.328i·37-s − 0.312·41-s + 0.609i·43-s + 1.16i·47-s − 1.28·49-s − 1.37i·53-s − 0.520·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.894 - 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{7200} (6049, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7200,\ (\ :1/2),\ 0.894 - 0.447i)\)
\(L(1)\)  \(\approx\)  \(1.710968545\)
\(L(\frac12)\)  \(\approx\)  \(1.710968545\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.030945274866290624391395932123, −7.41012102177614558582520485635, −6.49678300143192144470358518034, −5.75987944408407829034944734107, −5.10119823134211183999731131145, −4.79590770474619549983835209966, −3.24814948191104928071163887097, −2.82889786957105707272879037880, −2.10052385655208064023860102234, −0.68396449882496674321471492532, 0.63653679722137863716449576704, 1.62615832698289726340825412889, 2.68245769799263935496630030517, 3.67001684898835786902007793321, 4.16671601641457576222334528541, 5.01970219584623735485383762292, 5.72515634181555500663105018751, 6.67226656295563449905007386284, 7.13903553524199081565810877067, 7.942436900402086903034962248441

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