Properties

Label 2-4864-1.1-c1-0-114
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  − 2.63·3-s + 2.63·5-s + 3.88·7-s + 3.95·9-s − 5.95·11-s + 4.14·13-s − 6.94·15-s + 1.82·17-s + 19-s − 10.2·21-s − 4.14·23-s + 1.93·25-s − 2.50·27-s − 8.19·29-s + 1.67·31-s + 15.6·33-s + 10.2·35-s − 8.14·37-s − 10.9·39-s − 0.514·41-s − 10.9·43-s + 10.4·45-s − 13.3·47-s + 8.10·49-s − 4.82·51-s + 1.62·53-s − 15.6·55-s + ⋯
L(s)  = 1  − 1.52·3-s + 1.17·5-s + 1.46·7-s + 1.31·9-s − 1.79·11-s + 1.15·13-s − 1.79·15-s + 0.443·17-s + 0.229·19-s − 2.23·21-s − 0.865·23-s + 0.387·25-s − 0.482·27-s − 1.52·29-s + 0.301·31-s + 2.73·33-s + 1.73·35-s − 1.33·37-s − 1.75·39-s − 0.0803·41-s − 1.67·43-s + 1.55·45-s − 1.94·47-s + 1.15·49-s − 0.675·51-s + 0.223·53-s − 2.11·55-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 \( 1 + 2.63T + 3T^{2} \)
5 \( 1 - 2.63T + 5T^{2} \)
7 \( 1 - 3.88T + 7T^{2} \)
11 \( 1 + 5.95T + 11T^{2} \)
13 \( 1 - 4.14T + 13T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
23 \( 1 + 4.14T + 23T^{2} \)
29 \( 1 + 8.19T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 + 8.14T + 37T^{2} \)
41 \( 1 + 0.514T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 13.3T + 47T^{2} \)
53 \( 1 - 1.62T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 5.98T + 61T^{2} \)
67 \( 1 - 4.09T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 9.28T + 73T^{2} \)
79 \( 1 + 2.42T + 79T^{2} \)
83 \( 1 + 0.987T + 83T^{2} \)
89 \( 1 + 2.25T + 89T^{2} \)
97 \( 1 - 9.27T + 97T^{2} \)
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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

−7.986285516967496214335333658798, −7.06598196984867526208255172011, −6.12757286522740240472052892691, −5.63243785111437185225619431733, −5.18600623607968663291209305706, −4.65440202459396085442378487572, −3.34009640864432292590228163454, −1.94542682798523262379102156840, −1.47482018650289072896142198892, 0, 1.47482018650289072896142198892, 1.94542682798523262379102156840, 3.34009640864432292590228163454, 4.65440202459396085442378487572, 5.18600623607968663291209305706, 5.63243785111437185225619431733, 6.12757286522740240472052892691, 7.06598196984867526208255172011, 7.986285516967496214335333658798

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