Properties

Degree 4
Conductor $ 13^{2} \cdot 619^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 2·4-s + 2·5-s − 3·6-s − 3·7-s − 3·8-s + 2·9-s + 2·10-s + 10·11-s + 6·12-s − 2·13-s − 3·14-s − 6·15-s + 16-s + 4·17-s + 2·18-s − 12·19-s − 4·20-s + 9·21-s + 10·22-s − 8·23-s + 9·24-s − 2·25-s − 2·26-s + 6·27-s + 6·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 4-s + 0.894·5-s − 1.22·6-s − 1.13·7-s − 1.06·8-s + 2/3·9-s + 0.632·10-s + 3.01·11-s + 1.73·12-s − 0.554·13-s − 0.801·14-s − 1.54·15-s + 1/4·16-s + 0.970·17-s + 0.471·18-s − 2.75·19-s − 0.894·20-s + 1.96·21-s + 2.13·22-s − 1.66·23-s + 1.83·24-s − 2/5·25-s − 0.392·26-s + 1.15·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 64754209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 64754209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(64754209\)    =    \(13^{2} \cdot 619^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8047} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 64754209,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.138110062$
$L(\frac12)$  $\approx$  $1.138110062$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;619\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;619\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad13$C_1$ \( ( 1 + T )^{2} \)
619$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 12 T + 69 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 73 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 17 T + 155 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 11 T + 117 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 51 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 179 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 19 T + 253 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.160808142820660269042076563662, −7.51765719916662807588965463678, −6.78539019149311761263797141003, −6.70378590707603140027602997971, −6.53047568122544348573481725286, −6.18954150855916691256307382640, −5.85944248714376720987578440052, −5.72446600462784541316446902832, −5.32172201460189161230890811826, −4.94258001466775102999026884635, −4.25175162268397730553425788011, −4.20241811247347596179618538660, −4.03306415257982623498696513385, −3.59427422183073332827817802558, −2.99403507139816978495714122321, −2.51826035598364231014167048931, −1.71092222731916602807218821990, −1.65837434190666035108952209561, −0.58497649156770714201763802710, −0.44544847473242977688641858282, 0.44544847473242977688641858282, 0.58497649156770714201763802710, 1.65837434190666035108952209561, 1.71092222731916602807218821990, 2.51826035598364231014167048931, 2.99403507139816978495714122321, 3.59427422183073332827817802558, 4.03306415257982623498696513385, 4.20241811247347596179618538660, 4.25175162268397730553425788011, 4.94258001466775102999026884635, 5.32172201460189161230890811826, 5.72446600462784541316446902832, 5.85944248714376720987578440052, 6.18954150855916691256307382640, 6.53047568122544348573481725286, 6.70378590707603140027602997971, 6.78539019149311761263797141003, 7.51765719916662807588965463678, 8.160808142820660269042076563662

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