Properties

Degree $2$
Conductor $4719$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s − 12-s − 13-s − 2·15-s − 16-s + 6·17-s + 18-s + 4·19-s + 2·20-s − 8·23-s − 3·24-s − 25-s − 26-s + 27-s + 10·29-s − 2·30-s + 5·32-s + 6·34-s − 36-s + 6·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.277·13-s − 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.85·29-s − 0.365·30-s + 0.883·32-s + 1.02·34-s − 1/6·36-s + 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4719\)    =    \(3 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{4719} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4719,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
11 \( 1 \)
13 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.067240339264519261018751023838, −7.40954997679622783053664282350, −6.33619991158222726990578636227, −5.64452216048267524562070305362, −4.70735289538630101276893653791, −4.22762738057020770225993714933, −3.29815205297078749887366214518, −2.96482688583601030780106082781, −1.40441285520901475377581138398, 0, 1.40441285520901475377581138398, 2.96482688583601030780106082781, 3.29815205297078749887366214518, 4.22762738057020770225993714933, 4.70735289538630101276893653791, 5.64452216048267524562070305362, 6.33619991158222726990578636227, 7.40954997679622783053664282350, 8.067240339264519261018751023838

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