Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{2} \cdot 13 \cdot 17 $
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 − 4-s + 2·5-s + 3·8-s − 6·9-s − 2·10-s − 13-s − 16-s + 3·17-s + 6·18-s − 2·20-s + 3·25-s + 26-s − 4·29-s − 5·32-s − 3·34-s + 6·36-s − 4·37-s + 6·40-s − 12·41-s − 12·45-s + 2·49-s − 3·50-s + 52-s − 20·53-s + 4·58-s + 12·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 2·9-s − 0.632·10-s − 0.277·13-s − 1/4·16-s + 0.727·17-s + 1.41·18-s − 0.447·20-s + 3/5·25-s + 0.196·26-s − 0.742·29-s − 0.883·32-s − 0.514·34-s + 36-s − 0.657·37-s + 0.948·40-s − 1.87·41-s − 1.78·45-s + 2/7·49-s − 0.424·50-s + 0.138·52-s − 2.74·53-s + 0.525·58-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 88400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 88400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88400\)    =    \(2^{4} \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{88400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 88400,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;13,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.445370946078724095421771046114, −8.908301443936015740885982494409, −8.540017954936382733473658077211, −8.057484724505346957367765946867, −7.68505998628538599450969837444, −6.83549653004652567501187752741, −6.35657857212787094671023234155, −5.67610609191103462331322799059, −5.20829761143314977818908810247, −4.96837461124932327787635715000, −3.80580534244267015608673762747, −3.21221371525954144695929729012, −2.42012886642585914453369006792, −1.49982897493095869103804944828, 0, 1.49982897493095869103804944828, 2.42012886642585914453369006792, 3.21221371525954144695929729012, 3.80580534244267015608673762747, 4.96837461124932327787635715000, 5.20829761143314977818908810247, 5.67610609191103462331322799059, 6.35657857212787094671023234155, 6.83549653004652567501187752741, 7.68505998628538599450969837444, 8.057484724505346957367765946867, 8.540017954936382733473658077211, 8.908301443936015740885982494409, 9.445370946078724095421771046114

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