Properties

Label 4-799e2-1.1-c0e2-0-0
Degree $4$
Conductor $638401$
Sign $1$
Analytic cond. $0.159003$
Root an. cond. $0.631468$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s − 2·7-s + 2·9-s − 4·12-s + 3·16-s + 2·17-s − 4·21-s + 2·27-s + 4·28-s − 4·36-s − 2·37-s − 2·47-s + 6·48-s + 2·49-s + 4·51-s + 2·61-s − 4·63-s − 4·64-s − 4·68-s + 2·71-s + 2·79-s + 3·81-s + 8·84-s + 4·89-s − 2·97-s − 4·108-s + ⋯
L(s)  = 1  + 2·3-s − 2·4-s − 2·7-s + 2·9-s − 4·12-s + 3·16-s + 2·17-s − 4·21-s + 2·27-s + 4·28-s − 4·36-s − 2·37-s − 2·47-s + 6·48-s + 2·49-s + 4·51-s + 2·61-s − 4·63-s − 4·64-s − 4·68-s + 2·71-s + 2·79-s + 3·81-s + 8·84-s + 4·89-s − 2·97-s − 4·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 638401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 638401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(638401\)    =    \(17^{2} \cdot 47^{2}\)
Sign: $1$
Analytic conductor: \(0.159003\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 638401,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9106667539\)
\(L(\frac12)\) \(\approx\) \(0.9106667539\)
\(L(1)\) 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)$
bad17$C_1$ \( ( 1 - T )^{2} \)
47$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−10.30813916166489070980551840416, −10.02162990233791658777688878230, −9.594040985757889731963605525407, −9.385843929804427785077256885783, −9.149004211195580521395692797172, −8.517647138644809170150742433872, −8.386354777601009821054263059587, −7.80485911257533043226855360494, −7.67952333185749838503850381614, −6.70849618064936116204528590558, −6.56318741009394828702748829611, −5.74635104893792175643103972437, −5.08622314534893812886407372604, −5.00742648746650945834870031272, −3.86665528610201218482511266896, −3.71067770523075155581245407326, −3.20984728687798741128797199671, −3.19389671564216599440055642828, −2.16326964126628773584340625955, −0.974320373461786274110774279262, 0.974320373461786274110774279262, 2.16326964126628773584340625955, 3.19389671564216599440055642828, 3.20984728687798741128797199671, 3.71067770523075155581245407326, 3.86665528610201218482511266896, 5.00742648746650945834870031272, 5.08622314534893812886407372604, 5.74635104893792175643103972437, 6.56318741009394828702748829611, 6.70849618064936116204528590558, 7.67952333185749838503850381614, 7.80485911257533043226855360494, 8.386354777601009821054263059587, 8.517647138644809170150742433872, 9.149004211195580521395692797172, 9.385843929804427785077256885783, 9.594040985757889731963605525407, 10.02162990233791658777688878230, 10.30813916166489070980551840416

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