Properties

Degree 4
Conductor $ 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 5·9-s − 6·11-s − 4·16-s + 11·25-s + 10·29-s + 4·31-s + 4·41-s − 20·45-s − 49-s + 24·55-s − 20·59-s − 16·61-s − 16·71-s + 10·79-s + 16·80-s + 16·81-s − 30·99-s + 24·101-s − 10·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + ⋯
L(s)  = 1  − 1.78·5-s + 5/3·9-s − 1.80·11-s − 16-s + 11/5·25-s + 1.85·29-s + 0.718·31-s + 0.624·41-s − 2.98·45-s − 1/7·49-s + 3.23·55-s − 2.60·59-s − 2.04·61-s − 1.89·71-s + 1.12·79-s + 1.78·80-s + 16/9·81-s − 3.01·99-s + 2.38·101-s − 0.957·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1225\)    =    \(5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1225,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.4675386452\)
\(L(\frac12)\)  \(\approx\)  \(0.4675386452\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−13.73908249804327735744761172916, −13.34283107156104634236933164891, −12.39216276691738128396285050555, −12.36148837795374284937339794164, −11.42621865015752539898361587727, −10.57199601674620508871050040716, −10.42068121271810089671381063107, −9.316604718406395525588995501970, −8.382229318624263597580461235223, −7.69057201102210773213766625761, −7.35689965275207777678389254869, −6.41057499424661595037771355877, −4.60140633504418596338695192636, −4.58669105072248681251752024317, −3.02592800233480114290664971806, 3.02592800233480114290664971806, 4.58669105072248681251752024317, 4.60140633504418596338695192636, 6.41057499424661595037771355877, 7.35689965275207777678389254869, 7.69057201102210773213766625761, 8.382229318624263597580461235223, 9.316604718406395525588995501970, 10.42068121271810089671381063107, 10.57199601674620508871050040716, 11.42621865015752539898361587727, 12.36148837795374284937339794164, 12.39216276691738128396285050555, 13.34283107156104634236933164891, 13.73908249804327735744761172916

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