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Properties

Label 4-3744e2-1.1-c0e2-0-0
Degree $4$
Conductor $14017536$
Sign $1$
Analytic cond. $3.49129$
Root an. cond. $1.36693$
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  − 2·5-s + 2·25-s − 4·29-s + 2·37-s − 2·41-s − 4·61-s − 2·73-s + 2·89-s − 2·97-s − 2·109-s − 4·113-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s − 4·185-s + ⋯
L(s)  = 1  − 2·5-s + 2·25-s − 4·29-s + 2·37-s − 2·41-s − 4·61-s − 2·73-s + 2·89-s − 2·97-s − 2·109-s − 4·113-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s − 4·185-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14017536\)    =    \(2^{10} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3.49129\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14017536,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3185122488\)
\(L(\frac12)\) \(\approx\) \(0.3185122488\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
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$C_1$ \( ( 1 + T )^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$ \( ( 1 + T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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

−9.057932710982769147943482028470, −8.339742374672961279165232031769, −8.016814986215537215836916690020, −7.67500869256278450943271833205, −7.60081050113357637131959733652, −7.22151926179195700398412027215, −6.78302051541003112253828766859, −6.30573968906806554271417082287, −5.93036227048239425921511669264, −5.39546428376376608471313126149, −5.18708685007563075831132863235, −4.47937228504691894459332898697, −4.25487499357865804268424525686, −3.85431070512331280567216219998, −3.61202921888656863601696788705, −2.95321525711208065581558515371, −2.80431432439921721604670244259, −1.67573917283357882291012161368, −1.61508144026907618671371936935, −0.32072359653524332973986674631, 0.32072359653524332973986674631, 1.61508144026907618671371936935, 1.67573917283357882291012161368, 2.80431432439921721604670244259, 2.95321525711208065581558515371, 3.61202921888656863601696788705, 3.85431070512331280567216219998, 4.25487499357865804268424525686, 4.47937228504691894459332898697, 5.18708685007563075831132863235, 5.39546428376376608471313126149, 5.93036227048239425921511669264, 6.30573968906806554271417082287, 6.78302051541003112253828766859, 7.22151926179195700398412027215, 7.60081050113357637131959733652, 7.67500869256278450943271833205, 8.016814986215537215836916690020, 8.339742374672961279165232031769, 9.057932710982769147943482028470

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