Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{4} \cdot 11^{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  + 2·3-s + 4-s − 3·9-s − 3·11-s + 2·12-s + 16-s + 12·23-s − 14·27-s + 4·31-s − 6·33-s − 3·36-s + 4·37-s − 3·44-s + 24·47-s + 2·48-s − 10·49-s + 12·53-s + 64-s − 26·67-s + 24·69-s + 24·71-s − 4·81-s + 30·89-s + 12·92-s + 8·93-s + 4·97-s + 9·99-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 9-s − 0.904·11-s + 0.577·12-s + 1/4·16-s + 2.50·23-s − 2.69·27-s + 0.718·31-s − 1.04·33-s − 1/2·36-s + 0.657·37-s − 0.452·44-s + 3.50·47-s + 0.288·48-s − 1.42·49-s + 1.64·53-s + 1/8·64-s − 3.17·67-s + 2.88·69-s + 2.84·71-s − 4/9·81-s + 3.17·89-s + 1.25·92-s + 0.829·93-s + 0.406·97-s + 0.904·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(302500\)    =    \(2^{2} \cdot 5^{4} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{302500} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 302500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.576414584$
$L(\frac12)$  $\approx$  $2.576414584$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.942369524069980633833404820232, −8.316812877075225116243141130520, −8.003379760199122716602575955438, −7.37355750361837418705145948004, −7.29853134793764088505009214313, −6.39299024601606568879362228127, −6.05410683290230188214573869250, −5.25591445959428735491861815623, −5.20592148788576143134616479765, −4.25751151289549141315285174849, −3.54905277576412147109484999920, −2.96421704926700326951259240062, −2.65155040417235216240951299295, −2.18944587268591749591824641427, −0.870527168442070003633023384518, 0.870527168442070003633023384518, 2.18944587268591749591824641427, 2.65155040417235216240951299295, 2.96421704926700326951259240062, 3.54905277576412147109484999920, 4.25751151289549141315285174849, 5.20592148788576143134616479765, 5.25591445959428735491861815623, 6.05410683290230188214573869250, 6.39299024601606568879362228127, 7.29853134793764088505009214313, 7.37355750361837418705145948004, 8.003379760199122716602575955438, 8.316812877075225116243141130520, 8.942369524069980633833404820232

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