Properties

Degree 4
Conductor $ 5^{2} \cdot 29^{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  − 4-s − 2·7-s − 2·9-s − 8·13-s − 3·16-s − 6·23-s + 25-s + 2·28-s + 6·29-s + 2·36-s − 2·49-s + 8·52-s + 12·59-s + 4·63-s + 7·64-s + 10·67-s − 12·71-s − 5·81-s − 18·83-s + 16·91-s + 6·92-s − 100-s + 10·103-s − 30·107-s + 16·109-s + 6·112-s − 6·116-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s − 2/3·9-s − 2.21·13-s − 3/4·16-s − 1.25·23-s + 1/5·25-s + 0.377·28-s + 1.11·29-s + 1/3·36-s − 2/7·49-s + 1.10·52-s + 1.56·59-s + 0.503·63-s + 7/8·64-s + 1.22·67-s − 1.42·71-s − 5/9·81-s − 1.97·83-s + 1.67·91-s + 0.625·92-s − 0.0999·100-s + 0.985·103-s − 2.90·107-s + 1.53·109-s + 0.566·112-s − 0.557·116-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21025\)    =    \(5^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{21025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 21025,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
29$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−10.31021475504937948132104477448, −9.885987153651444884556648357228, −9.696396268952817799509713689061, −8.942926438758517872552270660149, −8.471425133280778904402131824110, −7.85912181863133156548032228725, −7.12722301011653113774252073602, −6.72933335047920367945929344948, −5.95537804679188128605938024551, −5.26242766216616259639501937046, −4.66787191806607434412184181990, −4.00198380042866898411922941051, −2.91417830661177268148435641391, −2.32769573124223567048228946598, 0, 2.32769573124223567048228946598, 2.91417830661177268148435641391, 4.00198380042866898411922941051, 4.66787191806607434412184181990, 5.26242766216616259639501937046, 5.95537804679188128605938024551, 6.72933335047920367945929344948, 7.12722301011653113774252073602, 7.85912181863133156548032228725, 8.471425133280778904402131824110, 8.942926438758517872552270660149, 9.696396268952817799509713689061, 9.885987153651444884556648357228, 10.31021475504937948132104477448

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