Properties

Label 2-4864-1.1-c1-0-135
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  + 1.70·3-s + 1.66·5-s − 1.99·7-s − 0.0874·9-s + 2.08·11-s − 4.77·13-s + 2.83·15-s + 2.10·17-s + 19-s − 3.40·21-s − 4.84·23-s − 2.23·25-s − 5.26·27-s + 0.695·29-s − 9.77·31-s + 3.56·33-s − 3.31·35-s − 0.0772·37-s − 8.14·39-s − 10.7·41-s − 1.43·43-s − 0.145·45-s + 2.88·47-s − 3.01·49-s + 3.59·51-s − 9.00·53-s + 3.46·55-s + ⋯
L(s)  = 1  + 0.985·3-s + 0.743·5-s − 0.754·7-s − 0.0291·9-s + 0.629·11-s − 1.32·13-s + 0.732·15-s + 0.510·17-s + 0.229·19-s − 0.743·21-s − 1.01·23-s − 0.447·25-s − 1.01·27-s + 0.129·29-s − 1.75·31-s + 0.620·33-s − 0.560·35-s − 0.0127·37-s − 1.30·39-s − 1.68·41-s − 0.219·43-s − 0.0216·45-s + 0.420·47-s − 0.431·49-s + 0.503·51-s − 1.23·53-s + 0.467·55-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 \( 1 - 1.70T + 3T^{2} \)
5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 + 1.99T + 7T^{2} \)
11 \( 1 - 2.08T + 11T^{2} \)
13 \( 1 + 4.77T + 13T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
23 \( 1 + 4.84T + 23T^{2} \)
29 \( 1 - 0.695T + 29T^{2} \)
31 \( 1 + 9.77T + 31T^{2} \)
37 \( 1 + 0.0772T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 - 2.88T + 47T^{2} \)
53 \( 1 + 9.00T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 1.27T + 67T^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 + 1.10T + 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−7.909642976736444562498043716035, −7.28289015571386536047683827298, −6.50449622853928351737151853595, −5.73590249239851759866166874860, −5.05973676745176366199048731764, −3.84294890947411396336223567631, −3.31239804138633269563935106091, −2.37168615200805705720609084950, −1.74703110755903597804330517541, 0, 1.74703110755903597804330517541, 2.37168615200805705720609084950, 3.31239804138633269563935106091, 3.84294890947411396336223567631, 5.05973676745176366199048731764, 5.73590249239851759866166874860, 6.50449622853928351737151853595, 7.28289015571386536047683827298, 7.909642976736444562498043716035

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