Properties

Label 2-4864-1.1-c1-0-80
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.37·3-s + 0.792·5-s + 3.31·7-s + 2.62·9-s + 3.37·11-s − 4.10·13-s + 1.87·15-s + 5·17-s + 19-s + 7.86·21-s + 2.52·23-s − 4.37·25-s − 0.883·27-s − 5.98·29-s + 3.46·31-s + 8·33-s + 2.62·35-s + 3.16·37-s − 9.74·39-s + 9.37·43-s + 2.08·45-s + 9.30·47-s + 4·49-s + 11.8·51-s − 9.45·53-s + 2.67·55-s + 2.37·57-s + ⋯
L(s)  = 1  + 1.36·3-s + 0.354·5-s + 1.25·7-s + 0.875·9-s + 1.01·11-s − 1.13·13-s + 0.485·15-s + 1.21·17-s + 0.229·19-s + 1.71·21-s + 0.526·23-s − 0.874·25-s − 0.169·27-s − 1.11·29-s + 0.622·31-s + 1.39·33-s + 0.444·35-s + 0.521·37-s − 1.56·39-s + 1.42·43-s + 0.310·45-s + 1.35·47-s + 0.571·49-s + 1.66·51-s − 1.29·53-s + 0.360·55-s + 0.314·57-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.369717702\)
\(L(\frac12)\) \(\approx\) \(4.369717702\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37T + 3T^{2} \)
5 \( 1 - 0.792T + 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 - 3.37T + 11T^{2} \)
13 \( 1 + 4.10T + 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 + 5.98T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 - 3.16T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 9.37T + 43T^{2} \)
47 \( 1 - 9.30T + 47T^{2} \)
53 \( 1 + 9.45T + 53T^{2} \)
59 \( 1 - 4.37T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 - 8.37T + 67T^{2} \)
71 \( 1 + 6.63T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 - 8.74T + 97T^{2} \)
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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

−8.220229721210940258780844571922, −7.60074937810421367325634216885, −7.27090664206300571037661990620, −6.01706482497047735954460234487, −5.29104997897496929759778826400, −4.40173542090580194111248683916, −3.70945113913475205592535777770, −2.75225953657021412235754978530, −2.01641601498408684273341264529, −1.19725689344034363301250183149, 1.19725689344034363301250183149, 2.01641601498408684273341264529, 2.75225953657021412235754978530, 3.70945113913475205592535777770, 4.40173542090580194111248683916, 5.29104997897496929759778826400, 6.01706482497047735954460234487, 7.27090664206300571037661990620, 7.60074937810421367325634216885, 8.220229721210940258780844571922

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