Properties

Label 4-357e2-1.1-c1e2-0-7
Degree $4$
Conductor $127449$
Sign $-1$
Analytic cond. $8.12625$
Root an. cond. $1.68838$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 4·7-s + 9-s − 6·11-s + 12·16-s + 18·23-s − 25-s + 16·28-s + 12·29-s − 4·36-s − 8·37-s − 14·43-s + 24·44-s + 9·49-s − 12·53-s − 4·63-s − 32·64-s − 8·67-s + 24·71-s + 24·77-s − 20·79-s + 81-s − 72·92-s − 6·99-s + 4·100-s + 18·107-s + 40·109-s + ⋯
L(s)  = 1  − 2·4-s − 1.51·7-s + 1/3·9-s − 1.80·11-s + 3·16-s + 3.75·23-s − 1/5·25-s + 3.02·28-s + 2.22·29-s − 2/3·36-s − 1.31·37-s − 2.13·43-s + 3.61·44-s + 9/7·49-s − 1.64·53-s − 0.503·63-s − 4·64-s − 0.977·67-s + 2.84·71-s + 2.73·77-s − 2.25·79-s + 1/9·81-s − 7.50·92-s − 0.603·99-s + 2/5·100-s + 1.74·107-s + 3.83·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(127449\)    =    \(3^{2} \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(8.12625\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 127449,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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

−9.139625451228424767056240959109, −8.618966621189813626434174913816, −8.522938517911553076706544737963, −7.76044884865200324165501143721, −7.22521844366863385910113943370, −6.66463575826783700250676263362, −6.13580656527827529767575928100, −5.14306131647545403565477997005, −5.04495215415329304235007921156, −4.76879607173120818228089465106, −3.73051779139000016834558062412, −3.01356233326132332503177574612, −3.00632648437536992753290178800, −1.06023509936170009525644393553, 0, 1.06023509936170009525644393553, 3.00632648437536992753290178800, 3.01356233326132332503177574612, 3.73051779139000016834558062412, 4.76879607173120818228089465106, 5.04495215415329304235007921156, 5.14306131647545403565477997005, 6.13580656527827529767575928100, 6.66463575826783700250676263362, 7.22521844366863385910113943370, 7.76044884865200324165501143721, 8.522938517911553076706544737963, 8.618966621189813626434174913816, 9.139625451228424767056240959109

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