Properties

Label 2-4864-1.1-c1-0-119
Degree $2$
Conductor $4864$
Sign $-1$
Analytic cond. $38.8392$
Root an. cond. $6.23211$
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  + 2.91·3-s − 3.91·5-s − 0.593·7-s + 5.51·9-s − 3.51·11-s + 2.51·13-s − 11.4·15-s + 3·17-s + 19-s − 1.73·21-s − 7.32·23-s + 10.3·25-s + 7.32·27-s + 5.73·29-s − 4.40·31-s − 10.2·33-s + 2.32·35-s − 3.42·37-s + 7.32·39-s − 10.2·41-s + 4.69·43-s − 21.5·45-s − 2.08·47-s − 6.64·49-s + 8.75·51-s + 1.08·53-s + 13.7·55-s + ⋯
L(s)  = 1  + 1.68·3-s − 1.75·5-s − 0.224·7-s + 1.83·9-s − 1.05·11-s + 0.696·13-s − 2.95·15-s + 0.727·17-s + 0.229·19-s − 0.377·21-s − 1.52·23-s + 2.06·25-s + 1.40·27-s + 1.06·29-s − 0.791·31-s − 1.78·33-s + 0.392·35-s − 0.563·37-s + 1.17·39-s − 1.59·41-s + 0.716·43-s − 3.21·45-s − 0.303·47-s − 0.949·49-s + 1.22·51-s + 0.148·53-s + 1.85·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4864\)    =    \(2^{8} \cdot 19\)
Sign: $-1$
Analytic conductor: \(38.8392\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 2.91T + 3T^{2} \)
5 \( 1 + 3.91T + 5T^{2} \)
7 \( 1 + 0.593T + 7T^{2} \)
11 \( 1 + 3.51T + 11T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 + 7.32T + 23T^{2} \)
29 \( 1 - 5.73T + 29T^{2} \)
31 \( 1 + 4.40T + 31T^{2} \)
37 \( 1 + 3.42T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + 2.08T + 47T^{2} \)
53 \( 1 - 1.08T + 53T^{2} \)
59 \( 1 - 8.51T + 59T^{2} \)
61 \( 1 - 7.10T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 + 4.40T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 0.241T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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

−8.025167741749607129285791158354, −7.50261209280217650590127462704, −6.89614232310629463891195233474, −5.66722364720753825888588838957, −4.56558493168873408945667951852, −3.90112573591178263433771893446, −3.30550561707999527829066387829, −2.76977739007235451866222606569, −1.53085185742982527054096673613, 0, 1.53085185742982527054096673613, 2.76977739007235451866222606569, 3.30550561707999527829066387829, 3.90112573591178263433771893446, 4.56558493168873408945667951852, 5.66722364720753825888588838957, 6.89614232310629463891195233474, 7.50261209280217650590127462704, 8.025167741749607129285791158354

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