Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s − 16-s − 4·71-s − 81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 4·11-s − 16-s − 4·71-s − 81-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1500625\)    =    \(5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{1225} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1500625,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.329397800$
$L(\frac12)$  $\approx$  $1.329397800$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7\}$, \(F_p(T)\) is a polynomial of degree 4. 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 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
11$C_1$ \( ( 1 - T )^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$ \( ( 1 + T )^{4} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.808525302105704979753952440467, −9.756135287439598217163611926691, −9.202369470821391974638568171348, −8.927396946595284034690335471848, −8.706641390372644014055676846940, −8.330780849397573352821918570733, −7.36289440792070998940633942859, −7.26581164406950127297146803457, −6.79887453799926084924138946176, −6.40576646711482175871253270842, −6.05061216867643441238787335706, −5.76606802056393615800514373081, −4.68700314514062367333961334014, −4.60542854362445325517668098305, −3.95316090989407502307061453701, −3.76402609560450591347004108138, −3.13566014032118967832990293892, −2.34541662112690771483706212326, −1.48184834938861756288084290103, −1.29939866974928397743549768394, 1.29939866974928397743549768394, 1.48184834938861756288084290103, 2.34541662112690771483706212326, 3.13566014032118967832990293892, 3.76402609560450591347004108138, 3.95316090989407502307061453701, 4.60542854362445325517668098305, 4.68700314514062367333961334014, 5.76606802056393615800514373081, 6.05061216867643441238787335706, 6.40576646711482175871253270842, 6.79887453799926084924138946176, 7.26581164406950127297146803457, 7.36289440792070998940633942859, 8.330780849397573352821918570733, 8.706641390372644014055676846940, 8.927396946595284034690335471848, 9.202369470821391974638568171348, 9.756135287439598217163611926691, 9.808525302105704979753952440467

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