Properties

Label 2-4864-1.1-c1-0-101
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.59·3-s + 1.59·5-s + 4.34·7-s + 3.74·9-s + 1.74·11-s + 0.748·13-s + 4.15·15-s + 3·17-s − 19-s + 11.2·21-s − 1.94·23-s − 2.44·25-s + 1.94·27-s − 7.28·29-s + 0.654·31-s + 4.54·33-s + 6.94·35-s + 3.84·37-s + 1.94·39-s + 4.54·41-s − 10.4·43-s + 5.98·45-s + 7.59·47-s + 11.8·49-s + 7.79·51-s + 6.59·53-s + 2.79·55-s + ⋯
L(s)  = 1  + 1.49·3-s + 0.714·5-s + 1.64·7-s + 1.24·9-s + 0.527·11-s + 0.207·13-s + 1.07·15-s + 0.727·17-s − 0.229·19-s + 2.46·21-s − 0.405·23-s − 0.489·25-s + 0.374·27-s − 1.35·29-s + 0.117·31-s + 0.790·33-s + 1.17·35-s + 0.632·37-s + 0.311·39-s + 0.709·41-s − 1.59·43-s + 0.892·45-s + 1.10·47-s + 1.69·49-s + 1.09·51-s + 0.906·53-s + 0.376·55-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: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.099103048\)
\(L(\frac12)\) \(\approx\) \(5.099103048\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.59T + 3T^{2} \)
5 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 - 4.34T + 7T^{2} \)
11 \( 1 - 1.74T + 11T^{2} \)
13 \( 1 - 0.748T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 + 1.94T + 23T^{2} \)
29 \( 1 + 7.28T + 29T^{2} \)
31 \( 1 - 0.654T + 31T^{2} \)
37 \( 1 - 3.84T + 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 7.59T + 47T^{2} \)
53 \( 1 - 6.59T + 53T^{2} \)
59 \( 1 + 6.74T + 59T^{2} \)
61 \( 1 - 9.09T + 61T^{2} \)
67 \( 1 - 7.40T + 67T^{2} \)
71 \( 1 + 1.69T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 - 0.654T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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

−8.448335410158830724836758831730, −7.66038957901182645050346317725, −7.20069999566721537778080870363, −5.95934334447129486968326001659, −5.38064907240580567594579993384, −4.31336086373903142737721962229, −3.78904506404028199971961752491, −2.69665760729017619018860259171, −1.91245321568771744698548853411, −1.37852415059533913297329194488, 1.37852415059533913297329194488, 1.91245321568771744698548853411, 2.69665760729017619018860259171, 3.78904506404028199971961752491, 4.31336086373903142737721962229, 5.38064907240580567594579993384, 5.95934334447129486968326001659, 7.20069999566721537778080870363, 7.66038957901182645050346317725, 8.448335410158830724836758831730

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