Properties

Label 4-273e2-1.1-c1e2-0-20
Degree $4$
Conductor $74529$
Sign $1$
Analytic cond. $4.75203$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 2·4-s + 7-s + 6·9-s + 6·12-s − 7·13-s + 7·19-s + 3·21-s − 5·25-s + 9·27-s + 2·28-s − 7·31-s + 12·36-s + 21·37-s − 21·39-s − 10·43-s − 6·49-s − 14·52-s + 21·57-s − 12·61-s + 6·63-s − 8·64-s − 21·67-s − 7·73-s − 15·75-s + 14·76-s − 13·79-s + ⋯
L(s)  = 1  + 1.73·3-s + 4-s + 0.377·7-s + 2·9-s + 1.73·12-s − 1.94·13-s + 1.60·19-s + 0.654·21-s − 25-s + 1.73·27-s + 0.377·28-s − 1.25·31-s + 2·36-s + 3.45·37-s − 3.36·39-s − 1.52·43-s − 6/7·49-s − 1.94·52-s + 2.78·57-s − 1.53·61-s + 0.755·63-s − 64-s − 2.56·67-s − 0.819·73-s − 1.73·75-s + 1.60·76-s − 1.46·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.75203\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.252041185\)
\(L(\frac12)\) \(\approx\) \(3.252041185\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_2$ \( 1 + 7 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.94646614137454396352629979764, −11.65496242618452171295814140877, −11.45295539594080733592142442171, −10.60668621886620810502034235683, −10.03361538651004428937512375350, −9.738880242290970064130700907986, −9.270818919308251311088690319125, −8.952072348664637585990870387160, −7.997008540921849858929910544936, −7.77395084408797116776956823434, −7.28343775246572175703810103926, −7.23276783045830538793381531072, −6.21393502990790940835217823366, −5.67650974607376112580475113096, −4.57239956789591955913348624517, −4.54147780566097017972001263234, −3.24985603416689115049053828135, −2.99803856267134099805939180767, −2.22688617377351818073348755434, −1.67886043423194948274307306892, 1.67886043423194948274307306892, 2.22688617377351818073348755434, 2.99803856267134099805939180767, 3.24985603416689115049053828135, 4.54147780566097017972001263234, 4.57239956789591955913348624517, 5.67650974607376112580475113096, 6.21393502990790940835217823366, 7.23276783045830538793381531072, 7.28343775246572175703810103926, 7.77395084408797116776956823434, 7.997008540921849858929910544936, 8.952072348664637585990870387160, 9.270818919308251311088690319125, 9.738880242290970064130700907986, 10.03361538651004428937512375350, 10.60668621886620810502034235683, 11.45295539594080733592142442171, 11.65496242618452171295814140877, 11.94646614137454396352629979764

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