Properties

Label 4-5796e2-1.1-c1e2-0-1
Degree $4$
Conductor $33593616$
Sign $1$
Analytic cond. $2141.95$
Root an. cond. $6.80303$
Motivic weight $1$
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  + 5-s − 2·7-s + 6·11-s − 13-s − 4·19-s + 2·23-s − 8·25-s + 8·29-s − 6·31-s − 2·35-s + 10·37-s + 6·41-s − 7·43-s + 14·47-s + 3·49-s + 3·53-s + 6·55-s + 17·59-s + 5·61-s − 65-s − 7·67-s + 7·71-s + 8·73-s − 12·77-s + 6·83-s + 9·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 1.80·11-s − 0.277·13-s − 0.917·19-s + 0.417·23-s − 8/5·25-s + 1.48·29-s − 1.07·31-s − 0.338·35-s + 1.64·37-s + 0.937·41-s − 1.06·43-s + 2.04·47-s + 3/7·49-s + 0.412·53-s + 0.809·55-s + 2.21·59-s + 0.640·61-s − 0.124·65-s − 0.855·67-s + 0.830·71-s + 0.936·73-s − 1.36·77-s + 0.658·83-s + 0.953·89-s + 0.209·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33593616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33593616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33593616\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2141.95\)
Root analytic conductor: \(6.80303\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33593616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.877391068\)
\(L(\frac12)\) \(\approx\) \(3.877391068\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 17 T + 189 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 117 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 7 T + 153 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 157 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 113 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−8.251085227317418494539049818840, −8.019869663907959982211528225933, −7.43763539432361394242654250639, −7.23504956966065202411745843267, −6.72141782362359993198264871261, −6.53321223720773139165057428001, −6.16326257762804670589160905583, −5.98811959682143940937268491733, −5.33546866402550345257446633866, −5.28604983352227562016277925963, −4.38831192486960473502742275337, −4.27125519440573059816496619225, −3.88443672459926241105274017623, −3.60467791695951807820240116994, −2.97372072573913305483302064347, −2.54808807299819027260777583290, −2.03402978391882401862834238524, −1.78443794178621808871467510212, −0.76202891807342699441491496494, −0.73532796445813936415379525243, 0.73532796445813936415379525243, 0.76202891807342699441491496494, 1.78443794178621808871467510212, 2.03402978391882401862834238524, 2.54808807299819027260777583290, 2.97372072573913305483302064347, 3.60467791695951807820240116994, 3.88443672459926241105274017623, 4.27125519440573059816496619225, 4.38831192486960473502742275337, 5.28604983352227562016277925963, 5.33546866402550345257446633866, 5.98811959682143940937268491733, 6.16326257762804670589160905583, 6.53321223720773139165057428001, 6.72141782362359993198264871261, 7.23504956966065202411745843267, 7.43763539432361394242654250639, 8.019869663907959982211528225933, 8.251085227317418494539049818840

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