Properties

Label 4-39e4-1.1-c1e2-0-13
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $147.507$
Root an. cond. $3.48500$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 3·16-s − 6·17-s − 12·23-s − 7·25-s − 6·29-s − 16·43-s − 14·49-s + 6·53-s + 2·61-s + 7·64-s + 6·68-s + 8·79-s + 12·92-s + 7·100-s − 6·101-s + 20·103-s − 12·107-s + 30·113-s + 6·116-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 3/4·16-s − 1.45·17-s − 2.50·23-s − 7/5·25-s − 1.11·29-s − 2.43·43-s − 2·49-s + 0.824·53-s + 0.256·61-s + 7/8·64-s + 0.727·68-s + 0.900·79-s + 1.25·92-s + 7/10·100-s − 0.597·101-s + 1.97·103-s − 1.16·107-s + 2.82·113-s + 0.557·116-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(147.507\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2313441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
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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

−9.090624732530601026310937141632, −9.041723559381136267824340917759, −8.387822619347924294499169090191, −8.163524128386339054488294127688, −7.75884460601616925072064410954, −7.33113071713810766905902237075, −6.70435361669459934445453654506, −6.44118590881199747061221580175, −6.06398394553867898190904869553, −5.58788361213582033437527275522, −4.90381927800315230984413086383, −4.79260304176934536619508344813, −3.97438009968865693535853776413, −3.91304747946918563289983767918, −3.36480744552396524026032289560, −2.39072565639573916612797932788, −2.07269884376571401821555132095, −1.59421684282812097897887678615, 0, 0, 1.59421684282812097897887678615, 2.07269884376571401821555132095, 2.39072565639573916612797932788, 3.36480744552396524026032289560, 3.91304747946918563289983767918, 3.97438009968865693535853776413, 4.79260304176934536619508344813, 4.90381927800315230984413086383, 5.58788361213582033437527275522, 6.06398394553867898190904869553, 6.44118590881199747061221580175, 6.70435361669459934445453654506, 7.33113071713810766905902237075, 7.75884460601616925072064410954, 8.163524128386339054488294127688, 8.387822619347924294499169090191, 9.041723559381136267824340917759, 9.090624732530601026310937141632

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