Properties

Label 4-525e2-1.1-c1e2-0-20
Degree $4$
Conductor $275625$
Sign $1$
Analytic cond. $17.5740$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s − 3·7-s + 8·8-s + 9-s + 4·11-s − 12·14-s − 4·16-s + 4·18-s + 16·22-s + 12·23-s − 24·28-s + 20·29-s − 32·32-s + 8·36-s + 4·37-s + 2·43-s + 32·44-s + 48·46-s + 2·49-s − 8·53-s − 24·56-s + 80·58-s − 3·63-s − 64·64-s − 6·67-s − 16·71-s + ⋯
L(s)  = 1  + 2.82·2-s + 4·4-s − 1.13·7-s + 2.82·8-s + 1/3·9-s + 1.20·11-s − 3.20·14-s − 16-s + 0.942·18-s + 3.41·22-s + 2.50·23-s − 4.53·28-s + 3.71·29-s − 5.65·32-s + 4/3·36-s + 0.657·37-s + 0.304·43-s + 4.82·44-s + 7.07·46-s + 2/7·49-s − 1.09·53-s − 3.20·56-s + 10.5·58-s − 0.377·63-s − 8·64-s − 0.733·67-s − 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(275625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.5740\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 275625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.426101789\)
\(L(\frac12)\) \(\approx\) \(7.426101789\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5 \( 1 \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.979440455637625961269202062502, −8.569123090874178563709672380540, −7.60923711989545913207621904275, −6.82044341445639010249085887854, −6.79375443883521821659151920822, −6.28354708317492806397521188452, −6.05569215012514529764032801313, −5.28452479922674377650498409826, −4.65405693802966717273237919028, −4.62309319523854591570832526017, −3.98196758650784752573254151818, −3.24466149791278329969031792942, −3.04023520942509040038527723076, −2.56863529572190940234034504682, −1.13430400006930951329029124705, 1.13430400006930951329029124705, 2.56863529572190940234034504682, 3.04023520942509040038527723076, 3.24466149791278329969031792942, 3.98196758650784752573254151818, 4.62309319523854591570832526017, 4.65405693802966717273237919028, 5.28452479922674377650498409826, 6.05569215012514529764032801313, 6.28354708317492806397521188452, 6.79375443883521821659151920822, 6.82044341445639010249085887854, 7.60923711989545913207621904275, 8.569123090874178563709672380540, 8.979440455637625961269202062502

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