Properties

Label 4-9408e2-1.1-c1e2-0-24
Degree $4$
Conductor $88510464$
Sign $1$
Analytic cond. $5643.50$
Root an. cond. $8.66736$
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  + 2·3-s + 3·9-s − 8·25-s + 4·27-s − 8·31-s − 8·37-s − 24·47-s − 20·53-s − 16·75-s + 5·81-s + 8·83-s − 16·93-s − 24·103-s + 8·109-s − 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 48·141-s + 149-s + 151-s + 157-s − 40·159-s + 163-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 8/5·25-s + 0.769·27-s − 1.43·31-s − 1.31·37-s − 3.50·47-s − 2.74·53-s − 1.84·75-s + 5/9·81-s + 0.878·83-s − 1.65·93-s − 2.36·103-s + 0.766·109-s − 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 3.17·159-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5643.50\)
Root analytic conductor: \(8.66736\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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

−7.47387458896252166902402646729, −7.39961299145506272372634150237, −6.82273205312770361074092063355, −6.67614092340468741133135433208, −6.13254126799236570809664307564, −5.98530660742761897174697736855, −5.32051910888537439176944341085, −5.15722395037922820715191160760, −4.63526592857681755901962206415, −4.45486381885996857575382657400, −3.76137653143114479779026594425, −3.63716339073710508564694694954, −3.17483802462248879729345800131, −3.07622635213107252059363881560, −2.19219440194281609499012603618, −2.10384094357741018328998779064, −1.52168097207756385183619815163, −1.32335561967512417535557346485, 0, 0, 1.32335561967512417535557346485, 1.52168097207756385183619815163, 2.10384094357741018328998779064, 2.19219440194281609499012603618, 3.07622635213107252059363881560, 3.17483802462248879729345800131, 3.63716339073710508564694694954, 3.76137653143114479779026594425, 4.45486381885996857575382657400, 4.63526592857681755901962206415, 5.15722395037922820715191160760, 5.32051910888537439176944341085, 5.98530660742761897174697736855, 6.13254126799236570809664307564, 6.67614092340468741133135433208, 6.82273205312770361074092063355, 7.39961299145506272372634150237, 7.47387458896252166902402646729

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