Properties

Label 2-4800-5.4-c1-0-27
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + 5i·7-s − 9-s + 6·11-s + 3i·13-s + 2i·17-s + 19-s + 5·21-s − 2i·23-s + i·27-s + 6·29-s + 3·31-s − 6i·33-s − 6i·37-s + 3·39-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.88i·7-s − 0.333·9-s + 1.80·11-s + 0.832i·13-s + 0.485i·17-s + 0.229·19-s + 1.09·21-s − 0.417i·23-s + 0.192i·27-s + 1.11·29-s + 0.538·31-s − 1.04i·33-s − 0.986i·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

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

\(L(1)\) \(\approx\) \(2.131440817\)
\(L(\frac12)\) \(\approx\) \(2.131440817\)
\(L(\frac{3}{2})\) 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 \( 1 - 5iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 - 8iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.584753092487217925910789315590, −7.73568343271214195814678494758, −6.73642010792931587839690470639, −6.26422861346898765050741812747, −5.76164771756570215256683818429, −4.73140944595475011944813448837, −3.90278830921214390507713179810, −2.81703981853751883824249755605, −2.07188570304636866346138555998, −1.21221986866717236464162381646, 0.66526899198926756244062975514, 1.42464396943373685245798694693, 3.10268610706440176264105835414, 3.62655754323216977496943798067, 4.43643856529949201010740593328, 4.89018878615758594764240389838, 6.18874262511404438246195101406, 6.65214468058777236027927195718, 7.45444409110125013809191357255, 8.070386165258264988331779367625

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