Properties

Label 2-4864-1.1-c1-0-11
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  − 3.37·3-s − 2.52·5-s − 3.31·7-s + 8.37·9-s − 2.37·11-s + 5.84·13-s + 8.51·15-s + 5·17-s + 19-s + 11.1·21-s − 0.792·23-s + 1.37·25-s − 18.1·27-s − 2.67·29-s + 3.46·31-s + 8·33-s + 8.37·35-s − 10.0·37-s − 19.6·39-s + 3.62·43-s − 21.1·45-s − 0.644·47-s + 4·49-s − 16.8·51-s − 6.13·53-s + 5.98·55-s − 3.37·57-s + ⋯
L(s)  = 1  − 1.94·3-s − 1.12·5-s − 1.25·7-s + 2.79·9-s − 0.715·11-s + 1.61·13-s + 2.19·15-s + 1.21·17-s + 0.229·19-s + 2.44·21-s − 0.165·23-s + 0.274·25-s − 3.48·27-s − 0.496·29-s + 0.622·31-s + 1.39·33-s + 1.41·35-s − 1.65·37-s − 3.15·39-s + 0.553·43-s − 3.15·45-s − 0.0940·47-s + 0.571·49-s − 2.36·51-s − 0.842·53-s + 0.807·55-s − 0.446·57-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: $\chi_{4864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3661732365\)
\(L(\frac12)\) \(\approx\) \(0.3661732365\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.37T + 3T^{2} \)
5 \( 1 + 2.52T + 5T^{2} \)
7 \( 1 + 3.31T + 7T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
13 \( 1 - 5.84T + 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 + 0.792T + 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 3.62T + 43T^{2} \)
47 \( 1 + 0.644T + 47T^{2} \)
53 \( 1 + 6.13T + 53T^{2} \)
59 \( 1 + 1.37T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 - 2.62T + 67T^{2} \)
71 \( 1 - 6.63T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + 0.294T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 + 2.74T + 97T^{2} \)
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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

−7.969223399779388992821431753246, −7.43407497589195111340805844814, −6.63744186590410518119378440444, −6.05614520359229195344591643471, −5.53466079044148550933049550384, −4.66908676966322106139864359951, −3.76563390017770004755189342212, −3.30056629624189201885352272193, −1.39894578006807982759603688663, −0.39889333752079966465880502631, 0.39889333752079966465880502631, 1.39894578006807982759603688663, 3.30056629624189201885352272193, 3.76563390017770004755189342212, 4.66908676966322106139864359951, 5.53466079044148550933049550384, 6.05614520359229195344591643471, 6.63744186590410518119378440444, 7.43407497589195111340805844814, 7.969223399779388992821431753246

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