Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \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  + 4-s − 2·9-s + 16-s + 4·19-s − 5·25-s − 12·29-s − 8·31-s − 2·36-s + 12·41-s + 49-s − 12·59-s + 16·61-s + 64-s + 4·76-s + 16·79-s − 5·81-s − 12·89-s − 5·100-s + 4·109-s − 12·116-s − 22·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + ⋯
L(s)  = 1  + 1/2·4-s − 2/3·9-s + 1/4·16-s + 0.917·19-s − 25-s − 2.22·29-s − 1.43·31-s − 1/3·36-s + 1.87·41-s + 1/7·49-s − 1.56·59-s + 2.04·61-s + 1/8·64-s + 0.458·76-s + 1.80·79-s − 5/9·81-s − 1.27·89-s − 1/2·100-s + 0.383·109-s − 1.11·116-s − 2·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8778167717$
$L(\frac12)$  $\approx$  $0.8778167717$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.30526099005271094386573698884, −11.46841286601878794911833201781, −11.23136141438460806904855801814, −10.74873683533699837976941407431, −9.765547119459919407856461234632, −9.395997320196623402140164716317, −8.758164081717971221129539824522, −7.71895664384636679391588711627, −7.57571100088867902110310233811, −6.66965130292041403123447657278, −5.59857578974307577358363282055, −5.57928681742950427486583645839, −4.11500666640753372808858013263, −3.28327332646103144207952017478, −2.07204529193989626181377807359, 2.07204529193989626181377807359, 3.28327332646103144207952017478, 4.11500666640753372808858013263, 5.57928681742950427486583645839, 5.59857578974307577358363282055, 6.66965130292041403123447657278, 7.57571100088867902110310233811, 7.71895664384636679391588711627, 8.758164081717971221129539824522, 9.395997320196623402140164716317, 9.765547119459919407856461234632, 10.74873683533699837976941407431, 11.23136141438460806904855801814, 11.46841286601878794911833201781, 12.30526099005271094386573698884

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