Properties

Label 2-4800-5.4-c1-0-65
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  i·3-s − 9-s + 4·11-s + 2i·13-s − 2i·17-s − 8·19-s + 4i·23-s + i·27-s − 6·29-s − 4i·33-s − 2i·37-s + 2·39-s − 6·41-s − 4i·43-s + 12i·47-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s − 0.485i·17-s − 1.83·19-s + 0.834i·23-s + 0.192i·27-s − 1.11·29-s − 0.696i·33-s − 0.328i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s + 1.75i·47-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: \(1\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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

−7.71032510569072262086488957415, −7.17783413321688312870870099919, −6.35615423258339646253996866730, −5.98817181650402084242120908660, −4.83210746621997969870832160177, −4.09788697308073685171324925749, −3.29011294053351677908585571686, −2.10101473529025980216811474636, −1.45990141441098031357572591635, 0, 1.49785612267003571358929907306, 2.51633575047117498215008641202, 3.59876710541155682093588445546, 4.16413160757843991694599718276, 4.87662642103056694520551797063, 5.93574274819714052762172486210, 6.36133176608047715519177047249, 7.20954408718917121186469512149, 8.148999417763595037242873834441

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