Properties

Label 2-4864-1.1-c1-0-134
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  + 3.13·3-s − 0.594·5-s − 3.48·7-s + 6.83·9-s − 4.83·11-s − 0.215·13-s − 1.86·15-s + 1.29·17-s + 19-s − 10.9·21-s + 4.52·23-s − 4.64·25-s + 12.0·27-s − 9.41·29-s + 1.22·31-s − 15.1·33-s + 2.07·35-s − 5.62·37-s − 0.676·39-s + 0.450·41-s + 0.794·43-s − 4.06·45-s − 12.1·47-s + 5.16·49-s + 4.06·51-s − 2.56·53-s + 2.87·55-s + ⋯
L(s)  = 1  + 1.81·3-s − 0.265·5-s − 1.31·7-s + 2.27·9-s − 1.45·11-s − 0.0597·13-s − 0.481·15-s + 0.314·17-s + 0.229·19-s − 2.38·21-s + 0.944·23-s − 0.929·25-s + 2.31·27-s − 1.74·29-s + 0.219·31-s − 2.63·33-s + 0.350·35-s − 0.924·37-s − 0.108·39-s + 0.0702·41-s + 0.121·43-s − 0.605·45-s − 1.77·47-s + 0.737·49-s + 0.569·51-s − 0.352·53-s + 0.387·55-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 \( 1 - 3.13T + 3T^{2} \)
5 \( 1 + 0.594T + 5T^{2} \)
7 \( 1 + 3.48T + 7T^{2} \)
11 \( 1 + 4.83T + 11T^{2} \)
13 \( 1 + 0.215T + 13T^{2} \)
17 \( 1 - 1.29T + 17T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 + 9.41T + 29T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 + 5.62T + 37T^{2} \)
41 \( 1 - 0.450T + 41T^{2} \)
43 \( 1 - 0.794T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 2.56T + 53T^{2} \)
59 \( 1 - 2.75T + 59T^{2} \)
61 \( 1 + 7.76T + 61T^{2} \)
67 \( 1 + 4.11T + 67T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 + 3.08T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 2.08T + 97T^{2} \)
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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

−7.85771874840515909456257200830, −7.47081590028984465476335114322, −6.76241079701766293673298558166, −5.72024583081971674201512733217, −4.79986320104781523762208227490, −3.72931181382863528202963076006, −3.24768476099019675280658918047, −2.67359820807535463040880148434, −1.72548033297162996357982665709, 0, 1.72548033297162996357982665709, 2.67359820807535463040880148434, 3.24768476099019675280658918047, 3.72931181382863528202963076006, 4.79986320104781523762208227490, 5.72024583081971674201512733217, 6.76241079701766293673298558166, 7.47081590028984465476335114322, 7.85771874840515909456257200830

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