Properties

Label 2-4864-1.1-c1-0-131
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  + 1.14·3-s + 3.34·5-s − 3.34·7-s − 1.68·9-s + 0.489·11-s − 2.29·13-s + 3.83·15-s + 0.196·17-s − 19-s − 3.83·21-s − 4·23-s + 6.17·25-s − 5.37·27-s − 6.12·29-s + 8.22·31-s + 0.560·33-s − 11.1·35-s − 3.83·37-s − 2.62·39-s + 4.12·41-s − 7.17·43-s − 5.63·45-s − 11.0·47-s + 4.17·49-s + 0.225·51-s + 4.81·53-s + 1.63·55-s + ⋯
L(s)  = 1  + 0.661·3-s + 1.49·5-s − 1.26·7-s − 0.561·9-s + 0.147·11-s − 0.635·13-s + 0.989·15-s + 0.0476·17-s − 0.229·19-s − 0.836·21-s − 0.834·23-s + 1.23·25-s − 1.03·27-s − 1.13·29-s + 1.47·31-s + 0.0976·33-s − 1.88·35-s − 0.630·37-s − 0.420·39-s + 0.644·41-s − 1.09·43-s − 0.840·45-s − 1.60·47-s + 0.596·49-s + 0.0315·51-s + 0.660·53-s + 0.220·55-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 \( 1 - 1.14T + 3T^{2} \)
5 \( 1 - 3.34T + 5T^{2} \)
7 \( 1 + 3.34T + 7T^{2} \)
11 \( 1 - 0.489T + 11T^{2} \)
13 \( 1 + 2.29T + 13T^{2} \)
17 \( 1 - 0.196T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 6.12T + 29T^{2} \)
31 \( 1 - 8.22T + 31T^{2} \)
37 \( 1 + 3.83T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 7.17T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 4.81T + 53T^{2} \)
59 \( 1 - 5.31T + 59T^{2} \)
61 \( 1 - 7.73T + 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 0.728T + 83T^{2} \)
89 \( 1 + 8.51T + 89T^{2} \)
97 \( 1 + 3.31T + 97T^{2} \)
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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

−8.054770292806145878080047614015, −7.04449365499455863350815444152, −6.38134605864978275464984751855, −5.84718451685793862339249720746, −5.15463798286564384494245440250, −3.98274181416164005302539877509, −3.06557433604310803316915703107, −2.50593828345182557582032960108, −1.67397896112874662574482739478, 0, 1.67397896112874662574482739478, 2.50593828345182557582032960108, 3.06557433604310803316915703107, 3.98274181416164005302539877509, 5.15463798286564384494245440250, 5.84718451685793862339249720746, 6.38134605864978275464984751855, 7.04449365499455863350815444152, 8.054770292806145878080047614015

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