Properties

Degree 4
Conductor $ 2^{4} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 4·7-s + 3·9-s + 3·11-s + 4·13-s + 3·15-s − 3·17-s + 19-s + 4·21-s − 3·23-s + 5·25-s − 8·27-s − 12·29-s + 7·31-s − 3·33-s + 12·35-s + 37-s − 4·39-s + 12·41-s − 8·43-s − 9·45-s + 9·47-s + 9·49-s + 3·51-s − 3·53-s − 9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 1.51·7-s + 9-s + 0.904·11-s + 1.10·13-s + 0.774·15-s − 0.727·17-s + 0.229·19-s + 0.872·21-s − 0.625·23-s + 25-s − 1.53·27-s − 2.22·29-s + 1.25·31-s − 0.522·33-s + 2.02·35-s + 0.164·37-s − 0.640·39-s + 1.87·41-s − 1.21·43-s − 1.34·45-s + 1.31·47-s + 9/7·49-s + 0.420·51-s − 0.412·53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(784\)    =    \(2^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{28} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 784,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.358217$
$L(\frac12)$  $\approx$  $0.358217$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−17.52539665179351686539607658141, −16.83240693466222331245255824153, −16.22593751048373275065867264340, −15.95920936655884041627259306018, −15.34293187880635901845287645069, −14.93563706892529383383121625487, −13.54814894465822888147983726342, −13.40005263692909520970142355282, −12.33996226575045532080751057895, −12.20439698718194328052721890579, −11.04513563684512874615408261657, −11.02460749959139304874986304333, −9.616510756305673496977057334962, −9.378231197034825499779622084931, −8.202084321555248126345415607740, −7.32269192359830248395197349962, −6.59765781472494882013944777804, −5.88053399239330996514645357185, −4.16290105624966053923484697414, −3.68820621838161426787695488127, 3.68820621838161426787695488127, 4.16290105624966053923484697414, 5.88053399239330996514645357185, 6.59765781472494882013944777804, 7.32269192359830248395197349962, 8.202084321555248126345415607740, 9.378231197034825499779622084931, 9.616510756305673496977057334962, 11.02460749959139304874986304333, 11.04513563684512874615408261657, 12.20439698718194328052721890579, 12.33996226575045532080751057895, 13.40005263692909520970142355282, 13.54814894465822888147983726342, 14.93563706892529383383121625487, 15.34293187880635901845287645069, 15.95920936655884041627259306018, 16.22593751048373275065867264340, 16.83240693466222331245255824153, 17.52539665179351686539607658141

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