Properties

Label 2-4864-1.1-c1-0-0
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  − 2.46·3-s − 3.22·5-s − 4.04·7-s + 3.09·9-s + 6.41·11-s − 0.815·13-s + 7.96·15-s − 2.62·17-s + 19-s + 9.97·21-s − 6.16·23-s + 5.41·25-s − 0.233·27-s − 3.85·29-s − 7.43·31-s − 15.8·33-s + 13.0·35-s − 5.13·37-s + 2.01·39-s − 11.5·41-s − 0.0837·43-s − 9.98·45-s − 3.99·47-s + 9.33·49-s + 6.46·51-s − 4.22·53-s − 20.6·55-s + ⋯
L(s)  = 1  − 1.42·3-s − 1.44·5-s − 1.52·7-s + 1.03·9-s + 1.93·11-s − 0.226·13-s + 2.05·15-s − 0.635·17-s + 0.229·19-s + 2.17·21-s − 1.28·23-s + 1.08·25-s − 0.0449·27-s − 0.716·29-s − 1.33·31-s − 2.75·33-s + 2.20·35-s − 0.844·37-s + 0.322·39-s − 1.79·41-s − 0.0127·43-s − 1.48·45-s − 0.582·47-s + 1.33·49-s + 0.905·51-s − 0.580·53-s − 2.78·55-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

\(L(1)\) \(\approx\) \(0.08233110803\)
\(L(\frac12)\) \(\approx\) \(0.08233110803\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 3T^{2} \)
5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 + 4.04T + 7T^{2} \)
11 \( 1 - 6.41T + 11T^{2} \)
13 \( 1 + 0.815T + 13T^{2} \)
17 \( 1 + 2.62T + 17T^{2} \)
23 \( 1 + 6.16T + 23T^{2} \)
29 \( 1 + 3.85T + 29T^{2} \)
31 \( 1 + 7.43T + 31T^{2} \)
37 \( 1 + 5.13T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 + 0.0837T + 43T^{2} \)
47 \( 1 + 3.99T + 47T^{2} \)
53 \( 1 + 4.22T + 53T^{2} \)
59 \( 1 - 8.53T + 59T^{2} \)
61 \( 1 + 5.72T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 + 7.25T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 6.51T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−8.244561147462645188947874725452, −7.09013597539222764245555661115, −6.86553616512684133445147915419, −6.21401709561476542868432516447, −5.49085861093813440047308301445, −4.40975617468582461797011428919, −3.84060599222112837855653774318, −3.29259227524752633890899502791, −1.57481534635387973164606873747, −0.17107425515265243503084408167, 0.17107425515265243503084408167, 1.57481534635387973164606873747, 3.29259227524752633890899502791, 3.84060599222112837855653774318, 4.40975617468582461797011428919, 5.49085861093813440047308301445, 6.21401709561476542868432516447, 6.86553616512684133445147915419, 7.09013597539222764245555661115, 8.244561147462645188947874725452

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