Properties

Label 2-4864-1.1-c1-0-133
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  + 1.15·3-s + 1.07·5-s + 0.238·7-s − 1.67·9-s + 2.84·11-s − 1.31·13-s + 1.23·15-s − 4.47·17-s − 19-s + 0.274·21-s + 1.94·23-s − 3.84·25-s − 5.38·27-s − 8.83·29-s − 6.64·31-s + 3.27·33-s + 0.256·35-s + 4.06·37-s − 1.51·39-s − 2.78·41-s + 1.21·43-s − 1.80·45-s − 10.2·47-s − 6.94·49-s − 5.15·51-s + 8.36·53-s + 3.05·55-s + ⋯
L(s)  = 1  + 0.664·3-s + 0.481·5-s + 0.0901·7-s − 0.557·9-s + 0.857·11-s − 0.364·13-s + 0.319·15-s − 1.08·17-s − 0.229·19-s + 0.0599·21-s + 0.405·23-s − 0.768·25-s − 1.03·27-s − 1.64·29-s − 1.19·31-s + 0.569·33-s + 0.0433·35-s + 0.668·37-s − 0.242·39-s − 0.434·41-s + 0.185·43-s − 0.268·45-s − 1.49·47-s − 0.991·49-s − 0.721·51-s + 1.14·53-s + 0.412·55-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 \( 1 - 1.15T + 3T^{2} \)
5 \( 1 - 1.07T + 5T^{2} \)
7 \( 1 - 0.238T + 7T^{2} \)
11 \( 1 - 2.84T + 11T^{2} \)
13 \( 1 + 1.31T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
23 \( 1 - 1.94T + 23T^{2} \)
29 \( 1 + 8.83T + 29T^{2} \)
31 \( 1 + 6.64T + 31T^{2} \)
37 \( 1 - 4.06T + 37T^{2} \)
41 \( 1 + 2.78T + 41T^{2} \)
43 \( 1 - 1.21T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 8.36T + 53T^{2} \)
59 \( 1 + 9.25T + 59T^{2} \)
61 \( 1 - 4.66T + 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 - 2.78T + 71T^{2} \)
73 \( 1 + 0.134T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 9.20T + 89T^{2} \)
97 \( 1 + 0.247T + 97T^{2} \)
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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

−7.962857908910403399957204590128, −7.25606560106997614023235993967, −6.44610313283125018610110752250, −5.78954539014891677453167620444, −4.96933771150360799676890170128, −4.00299042290785335255413826516, −3.33566021420691983778385486717, −2.27831310397930921333534679448, −1.70979024365260129192829441671, 0, 1.70979024365260129192829441671, 2.27831310397930921333534679448, 3.33566021420691983778385486717, 4.00299042290785335255413826516, 4.96933771150360799676890170128, 5.78954539014891677453167620444, 6.44610313283125018610110752250, 7.25606560106997614023235993967, 7.962857908910403399957204590128

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