Properties

Degree 4
Conductor $ 5^{2} \cdot 61^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 3·5-s − 2·9-s − 10·11-s + 5·16-s − 8·19-s + 9·20-s + 4·25-s − 12·29-s + 6·36-s + 10·41-s + 30·44-s + 6·45-s − 13·49-s + 30·55-s + 18·59-s − 2·61-s − 3·64-s − 16·71-s + 24·76-s + 6·79-s − 15·80-s − 5·81-s − 8·89-s + 24·95-s + 20·99-s − 12·100-s + ⋯
L(s)  = 1  − 3/2·4-s − 1.34·5-s − 2/3·9-s − 3.01·11-s + 5/4·16-s − 1.83·19-s + 2.01·20-s + 4/5·25-s − 2.22·29-s + 36-s + 1.56·41-s + 4.52·44-s + 0.894·45-s − 1.85·49-s + 4.04·55-s + 2.34·59-s − 0.256·61-s − 3/8·64-s − 1.89·71-s + 2.75·76-s + 0.675·79-s − 1.67·80-s − 5/9·81-s − 0.847·89-s + 2.46·95-s + 2.01·99-s − 6/5·100-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 93025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 93025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(93025\)    =    \(5^{2} \cdot 61^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{93025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 93025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;61\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;61\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ \( 1 + 3 T + p T^{2} \)
61$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.019416501615119066110839147797, −8.400937388344254657063163947760, −8.287570693733701613432166498970, −7.66039763333771123227596439320, −7.58590694377702174449965560919, −6.61981736464800576569297638001, −5.60157612088078313975836309422, −5.50181452599748063455305172341, −4.82603389209875681234034335898, −4.25666915587159511841100490901, −3.82229793578931859480218288321, −2.97340716320340968167510315654, −2.30723040854003748390037888337, 0, 0, 2.30723040854003748390037888337, 2.97340716320340968167510315654, 3.82229793578931859480218288321, 4.25666915587159511841100490901, 4.82603389209875681234034335898, 5.50181452599748063455305172341, 5.60157612088078313975836309422, 6.61981736464800576569297638001, 7.58590694377702174449965560919, 7.66039763333771123227596439320, 8.287570693733701613432166498970, 8.400937388344254657063163947760, 9.019416501615119066110839147797

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