Properties

Label 2-4864-1.1-c1-0-120
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

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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: \(1\)
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 \( 1 - 2.59T + 3T^{2} \)
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 \( 1 + 10.5T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.032634514644768639886726751395, −7.26466243447047531584872378269, −6.68707647554705744889532641085, −5.91436476571299759613824884677, −4.64529720588814710557352430744, −3.82908969053331043801453637944, −3.20557504398427568596908214701, −2.83087175764492009134903730373, −1.53270217049283405653508955855, 0, 1.53270217049283405653508955855, 2.83087175764492009134903730373, 3.20557504398427568596908214701, 3.82908969053331043801453637944, 4.64529720588814710557352430744, 5.91436476571299759613824884677, 6.68707647554705744889532641085, 7.26466243447047531584872378269, 8.032634514644768639886726751395

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