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Properties

Label 4-1476e2-1.1-c0e2-0-4
Degree $4$
Conductor $2178576$
Sign $1$
Analytic cond. $0.542608$
Root an. cond. $0.858265$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·25-s + 6·32-s − 2·41-s − 4·50-s − 4·61-s + 7·64-s − 4·82-s − 6·100-s − 8·122-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + 167-s + 2·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·25-s + 6·32-s − 2·41-s − 4·50-s − 4·61-s + 7·64-s − 4·82-s − 6·100-s − 8·122-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + 167-s + 2·169-s + 173-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2178576\)    =    \(2^{4} \cdot 3^{4} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.542608\)
Root analytic conductor: \(0.858265\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2178576,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.249411902\)
\(L(\frac12)\) \(\approx\) \(4.249411902\)
\(L(1)\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
41$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 + T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.01421226632330594889713470662, −9.705968290597660400750604545000, −9.136129680727659168291888519768, −8.576248010243203570525885907267, −7.956604582039022768788725691215, −7.82329451436062946474700126531, −7.32973813055898425318770593469, −6.97002368675921360142881872629, −6.35259995173532438525362902785, −6.21390237571841117597290264831, −5.69905878489709890040044807750, −5.36805907639702349566274186492, −4.74543900634871059464864485533, −4.55830788875088394314565672338, −3.91819557144235609080736023061, −3.57154273321310064218252119978, −3.08045646205651256833887920390, −2.58434719491712651022413391077, −1.80455648630118609834177336532, −1.57076904818596277278193059180, 1.57076904818596277278193059180, 1.80455648630118609834177336532, 2.58434719491712651022413391077, 3.08045646205651256833887920390, 3.57154273321310064218252119978, 3.91819557144235609080736023061, 4.55830788875088394314565672338, 4.74543900634871059464864485533, 5.36805907639702349566274186492, 5.69905878489709890040044807750, 6.21390237571841117597290264831, 6.35259995173532438525362902785, 6.97002368675921360142881872629, 7.32973813055898425318770593469, 7.82329451436062946474700126531, 7.956604582039022768788725691215, 8.576248010243203570525885907267, 9.136129680727659168291888519768, 9.705968290597660400750604545000, 10.01421226632330594889713470662

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