Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5^{2} $
Sign $0.447 - 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 3i·7-s − 9-s + 2·11-s + 3i·13-s − 6i·17-s + 7·19-s − 3·21-s − 6i·23-s i·27-s − 2·29-s + 5·31-s + 2i·33-s + 10i·37-s − 3·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.13i·7-s − 0.333·9-s + 0.603·11-s + 0.832i·13-s − 1.45i·17-s + 1.60·19-s − 0.654·21-s − 1.25i·23-s − 0.192i·27-s − 0.371·29-s + 0.898·31-s + 0.348i·33-s + 1.64i·37-s − 0.480·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{4800} (3649, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4800,\ (\ :1/2),\ 0.447 - 0.894i)\)
\(L(1)\)  \(\approx\)  \(2.155247061\)
\(L(\frac12)\)  \(\approx\)  \(2.155247061\)
\(L(\frac{3}{2})\)   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 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 3iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−8.631404241044679861006796995703, −7.74695374493879989485486476684, −6.85219791896510048477784859271, −6.24483117204824828233149814307, −5.30571021535095644880219082614, −4.84443366446912856076163666733, −3.94374071108469997562821317102, −2.93671936319750153627805690969, −2.34322944440169654584417784809, −0.929107553801735562031206855379, 0.809929797341450769054964834201, 1.45423069952354415473069150467, 2.73845633439929779830144635468, 3.72202399456383203643213891536, 4.16798076828796999795368539329, 5.53250771010055608889464874269, 5.84185834318970742589768883747, 6.93174011971981996490514855232, 7.43140856264504493673792403403, 7.938221310560828292654243092291

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