Properties

Label 2-3072-1.1-c1-0-54
Degree $2$
Conductor $3072$
Sign $-1$
Analytic cond. $24.5300$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.47·5-s + 2.55·7-s + 9-s − 0.669·11-s − 4.08·13-s − 2.47·15-s + 6.44·17-s − 6.44·19-s + 2.55·21-s − 2.82·23-s + 1.11·25-s + 27-s − 4.35·29-s − 6.55·31-s − 0.669·33-s − 6.32·35-s + 3.85·37-s − 4.08·39-s + 0.788·41-s + 0.550·43-s − 2.47·45-s + 2.82·47-s − 0.458·49-s + 6.44·51-s − 3.64·53-s + 1.65·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.10·5-s + 0.966·7-s + 0.333·9-s − 0.201·11-s − 1.13·13-s − 0.638·15-s + 1.56·17-s − 1.47·19-s + 0.558·21-s − 0.589·23-s + 0.223·25-s + 0.192·27-s − 0.808·29-s − 1.17·31-s − 0.116·33-s − 1.06·35-s + 0.634·37-s − 0.653·39-s + 0.123·41-s + 0.0840·43-s − 0.368·45-s + 0.412·47-s − 0.0654·49-s + 0.902·51-s − 0.500·53-s + 0.223·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $-1$
Analytic conductor: \(24.5300\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
good5 \( 1 + 2.47T + 5T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 + 0.669T + 11T^{2} \)
13 \( 1 + 4.08T + 13T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + 6.44T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 4.35T + 29T^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 - 3.85T + 37T^{2} \)
41 \( 1 - 0.788T + 41T^{2} \)
43 \( 1 - 0.550T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 3.64T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 + 6.20T + 61T^{2} \)
67 \( 1 - 2.99T + 67T^{2} \)
71 \( 1 + 5.11T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 - 0.907T + 83T^{2} \)
89 \( 1 + 6.31T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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

−8.172629288751591139836357825541, −7.61021065636406481649947198260, −7.32114001069686960686272430081, −5.99585675755646830143526151690, −5.08522592817670779426477426363, −4.29437869352557165296074121033, −3.66694552812741968612395152101, −2.58821805959736701277353821831, −1.61313102755982744258902585133, 0, 1.61313102755982744258902585133, 2.58821805959736701277353821831, 3.66694552812741968612395152101, 4.29437869352557165296074121033, 5.08522592817670779426477426363, 5.99585675755646830143526151690, 7.32114001069686960686272430081, 7.61021065636406481649947198260, 8.172629288751591139836357825541

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