Properties

Label 2-4800-5.4-c1-0-60
Degree $2$
Conductor $4800$
Sign $-0.894 + 0.447i$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s − 9-s − 2i·13-s − 6i·17-s + 4·19-s − 8i·23-s + i·27-s − 2·29-s − 4·31-s + 10i·37-s − 2·39-s + 2·41-s − 4i·43-s + 8i·47-s + 7·49-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s − 0.554i·13-s − 1.45i·17-s + 0.917·19-s − 1.66i·23-s + 0.192i·27-s − 0.371·29-s − 0.718·31-s + 1.64i·37-s − 0.320·39-s + 0.312·41-s − 0.609i·43-s + 1.16i·47-s + 49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.200021513\)
\(L(\frac12)\) \(\approx\) \(1.200021513\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.78836093625334776465573551175, −7.34548609825308846561796589929, −6.60784960757236337997500695256, −5.82004934949027835210529867676, −5.08651605796836415823035017968, −4.33034270936793254315718176937, −3.09088517238200020554199613429, −2.62711201163926188229327463379, −1.36090735493308643597599855026, −0.33542539082138709577253890089, 1.36201600088555421932110646520, 2.31624474995359990651049566964, 3.66291170833758050763207099438, 3.79719581801783884373091038081, 4.97780553932737593400022108817, 5.63527264948320825461346414328, 6.25449061157180928154606929417, 7.34554876111416156679978408542, 7.73883503057465455724837766867, 8.810693558670117024080883748727

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