Properties

Label 4-2268e2-1.1-c1e2-0-22
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $327.974$
Root an. cond. $4.25559$
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  − 3·5-s − 7-s − 3·11-s − 2·13-s − 12·17-s + 10·19-s + 9·23-s + 5·25-s − 6·29-s + 31-s + 3·35-s + 22·37-s − 3·41-s + 4·43-s + 12·47-s + 9·55-s − 8·61-s + 6·65-s + 10·67-s + 6·71-s + 16·73-s + 3·77-s + 4·79-s − 6·83-s + 36·85-s + 6·89-s + 2·91-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.554·13-s − 2.91·17-s + 2.29·19-s + 1.87·23-s + 25-s − 1.11·29-s + 0.179·31-s + 0.507·35-s + 3.61·37-s − 0.468·41-s + 0.609·43-s + 1.75·47-s + 1.21·55-s − 1.02·61-s + 0.744·65-s + 1.22·67-s + 0.712·71-s + 1.87·73-s + 0.341·77-s + 0.450·79-s − 0.658·83-s + 3.90·85-s + 0.635·89-s + 0.209·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(327.974\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444699130\)
\(L(\frac12)\) \(\approx\) \(1.444699130\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−9.109250960128848972997426134932, −9.035466248044116569284691589345, −8.359479347050842524759564926339, −7.989497642399884095119267158415, −7.48754641558964616669323275404, −7.47895722820427653153758076706, −7.03144586320956580493068609299, −6.60691863899895351603798255049, −6.16382534142592603294555716637, −5.61790265268361645543184509712, −5.11271744289175224642211037923, −4.81553660681730190763099258868, −4.34233099286251036442013279753, −4.08409176059270317567287778000, −3.35452747723587546639411006469, −3.04576382091357396493691526387, −2.45927735731758332190150388725, −2.19068902211059161625229404736, −0.876616966665843496853661752538, −0.55865990341771668585250083067, 0.55865990341771668585250083067, 0.876616966665843496853661752538, 2.19068902211059161625229404736, 2.45927735731758332190150388725, 3.04576382091357396493691526387, 3.35452747723587546639411006469, 4.08409176059270317567287778000, 4.34233099286251036442013279753, 4.81553660681730190763099258868, 5.11271744289175224642211037923, 5.61790265268361645543184509712, 6.16382534142592603294555716637, 6.60691863899895351603798255049, 7.03144586320956580493068609299, 7.47895722820427653153758076706, 7.48754641558964616669323275404, 7.989497642399884095119267158415, 8.359479347050842524759564926339, 9.035466248044116569284691589345, 9.109250960128848972997426134932

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