Properties

Label 2-4800-5.4-c1-0-44
Degree $2$
Conductor $4800$
Sign $0.447 + 0.894i$
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 + i·7-s − 9-s + 6·11-s + 5i·13-s − 6i·17-s − 5·19-s + 21-s − 6i·23-s + i·27-s − 6·29-s + 31-s − 6i·33-s − 2i·37-s + 5·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 1.80·11-s + 1.38i·13-s − 1.45i·17-s − 1.14·19-s + 0.218·21-s − 1.25i·23-s + 0.192i·27-s − 1.11·29-s + 0.179·31-s − 1.04i·33-s − 0.328i·37-s + 0.800·39-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

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
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: \((2,\ 4800,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.918517580\)
\(L(\frac12)\) \(\approx\) \(1.918517580\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−8.327226487992534623057315491699, −7.05039447424064324140924532244, −6.85818785654417770486200675013, −6.22039407250702694528961799349, −5.25983896758101428045009510703, −4.30564465930205384807314079011, −3.75105380915317385580510289194, −2.42766151147122454065430182708, −1.84873406468707104252775814054, −0.61784650159165587218007995658, 0.998007505833307047385853231120, 2.01728678140894120673065218438, 3.39873202620584695147564440633, 3.82997973735432705378852402607, 4.50940263472266057808710524758, 5.65478699821192405326711161179, 6.07951121888303215833571830546, 6.92172840405867299956095063243, 7.79269256947370971533460886624, 8.452293957893836008865088767680

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