Properties

Label 4-39e4-1.1-c1e2-0-11
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 12·16-s − 12·23-s + 2·25-s − 12·29-s + 2·43-s − 11·49-s − 24·53-s + 2·61-s − 32·64-s − 22·79-s + 48·92-s − 8·100-s − 36·101-s + 2·103-s + 12·107-s − 12·113-s + 48·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 2.50·23-s + 2/5·25-s − 2.22·29-s + 0.304·43-s − 1.57·49-s − 3.29·53-s + 0.256·61-s − 4·64-s − 2.47·79-s + 5.00·92-s − 4/5·100-s − 3.58·101-s + 0.197·103-s + 1.16·107-s − 1.12·113-s + 4.45·116-s − 0.909·121-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 + 0.0798·157-s + 0.0783·163-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$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 + 167 T^{2} + p^{2} T^{4} \)
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

−9.263350720538986503949105687311, −9.023759865092393766925130882852, −8.433072077157405642270976195742, −8.050875138516453875748684461194, −7.86295233529838917236644390129, −7.56773026814495758560850966451, −6.78191667369722655303971926722, −6.31271200054070109359894624843, −5.78042354082766657034611225905, −5.58955418777624952506230146765, −5.04981090419579758935387031604, −4.62565579207585428587006442781, −4.01314425454951976637895234458, −4.00306439752385457084057769673, −3.34122237006366957014266780101, −2.82072525284652420726382253881, −1.77572593776434704193686502476, −1.41433462402830073203810317989, 0, 0, 1.41433462402830073203810317989, 1.77572593776434704193686502476, 2.82072525284652420726382253881, 3.34122237006366957014266780101, 4.00306439752385457084057769673, 4.01314425454951976637895234458, 4.62565579207585428587006442781, 5.04981090419579758935387031604, 5.58955418777624952506230146765, 5.78042354082766657034611225905, 6.31271200054070109359894624843, 6.78191667369722655303971926722, 7.56773026814495758560850966451, 7.86295233529838917236644390129, 8.050875138516453875748684461194, 8.433072077157405642270976195742, 9.023759865092393766925130882852, 9.263350720538986503949105687311

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