Properties

Degree 4
Conductor $ 2^{3} \cdot 5^{2} \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 4·7-s − 9-s − 4·11-s + 8·14-s − 4·16-s + 2·18-s + 8·22-s − 12·23-s − 25-s − 8·28-s + 6·29-s + 8·32-s − 2·36-s − 4·37-s − 4·43-s − 8·44-s + 24·46-s + 9·49-s + 2·50-s − 4·53-s − 12·58-s + 4·63-s − 8·64-s − 8·71-s + 8·74-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 1/3·9-s − 1.20·11-s + 2.13·14-s − 16-s + 0.471·18-s + 1.70·22-s − 2.50·23-s − 1/5·25-s − 1.51·28-s + 1.11·29-s + 1.41·32-s − 1/3·36-s − 0.657·37-s − 0.609·43-s − 1.20·44-s + 3.53·46-s + 9/7·49-s + 0.282·50-s − 0.549·53-s − 1.57·58-s + 0.503·63-s − 64-s − 0.949·71-s + 0.929·74-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9800\)    =    \(2^{3} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 9800,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$V_4$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
19$V_4$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$V_4$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$V_4$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$V_4$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
61$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$V_4$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 11 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$V_4$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
97$V_4$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
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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

−10.96097672855471999670455572731, −10.37607513004593898486559467056, −10.12237329174001403401743893851, −9.655994513335251748988791092002, −9.091765282848374607806661552858, −8.310228988411345187569079346461, −8.033829362621111138391851747125, −7.35406334022305023920247013189, −6.55879439557055451894355331192, −6.12675302007344876028355886290, −5.24187428187498459647629163439, −4.16888467428457029225152009207, −3.13497375947079764545470445617, −2.18467840969798473312579053168, 0, 2.18467840969798473312579053168, 3.13497375947079764545470445617, 4.16888467428457029225152009207, 5.24187428187498459647629163439, 6.12675302007344876028355886290, 6.55879439557055451894355331192, 7.35406334022305023920247013189, 8.033829362621111138391851747125, 8.310228988411345187569079346461, 9.091765282848374607806661552858, 9.655994513335251748988791092002, 10.12237329174001403401743893851, 10.37607513004593898486559467056, 10.96097672855471999670455572731

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