Properties

Label 2-4864-1.1-c1-0-125
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  + 2.24·3-s − 1.47·5-s − 1.47·7-s + 2.05·9-s + 4.77·11-s − 4.49·13-s − 3.30·15-s + 1.71·17-s + 19-s − 3.30·21-s − 4·23-s − 2.83·25-s − 2.11·27-s − 7.80·29-s + 4.13·31-s + 10.7·33-s + 2.16·35-s − 3.30·37-s − 10.1·39-s − 9.80·41-s − 1.83·43-s − 3.02·45-s + 5.14·47-s − 4.83·49-s + 3.86·51-s + 12.8·53-s − 7.02·55-s + ⋯
L(s)  = 1  + 1.29·3-s − 0.657·5-s − 0.555·7-s + 0.686·9-s + 1.44·11-s − 1.24·13-s − 0.854·15-s + 0.417·17-s + 0.229·19-s − 0.721·21-s − 0.834·23-s − 0.567·25-s − 0.407·27-s − 1.44·29-s + 0.742·31-s + 1.87·33-s + 0.365·35-s − 0.543·37-s − 1.62·39-s − 1.53·41-s − 0.280·43-s − 0.451·45-s + 0.750·47-s − 0.691·49-s + 0.541·51-s + 1.76·53-s − 0.947·55-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 \( 1 - 2.24T + 3T^{2} \)
5 \( 1 + 1.47T + 5T^{2} \)
7 \( 1 + 1.47T + 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 + 4.49T + 13T^{2} \)
17 \( 1 - 1.71T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 4.13T + 31T^{2} \)
37 \( 1 + 3.30T + 37T^{2} \)
41 \( 1 + 9.80T + 41T^{2} \)
43 \( 1 + 1.83T + 43T^{2} \)
47 \( 1 - 5.14T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 + 9.05T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 + 5.92T + 67T^{2} \)
71 \( 1 - 6.05T + 71T^{2} \)
73 \( 1 + 0.335T + 73T^{2} \)
79 \( 1 - 0.366T + 79T^{2} \)
83 \( 1 - 18.0T + 83T^{2} \)
89 \( 1 - 2.36T + 89T^{2} \)
97 \( 1 + 7.05T + 97T^{2} \)
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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

−7.86077112719659031423192267719, −7.41523289040994427804144317935, −6.67557874845470258146178642500, −5.79901475292111110312918767390, −4.72358051663273923894233348380, −3.74120306229997364821711344697, −3.54124165814501002870000558131, −2.49637005079455046753922161169, −1.61902924040344362616824778446, 0, 1.61902924040344362616824778446, 2.49637005079455046753922161169, 3.54124165814501002870000558131, 3.74120306229997364821711344697, 4.72358051663273923894233348380, 5.79901475292111110312918767390, 6.67557874845470258146178642500, 7.41523289040994427804144317935, 7.86077112719659031423192267719

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