Properties

Label 4-1344e2-1.1-c1e2-0-57
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
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-s + 3·5-s + 5·7-s + 3·11-s + 8·13-s + 3·15-s − 4·19-s + 5·21-s + 5·25-s − 27-s − 18·29-s + 31-s + 3·33-s + 15·35-s + 8·37-s + 8·39-s + 20·43-s + 6·47-s + 18·49-s − 3·53-s + 9·55-s − 4·57-s + 3·59-s − 10·61-s + 24·65-s − 10·67-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1.88·7-s + 0.904·11-s + 2.21·13-s + 0.774·15-s − 0.917·19-s + 1.09·21-s + 25-s − 0.192·27-s − 3.34·29-s + 0.179·31-s + 0.522·33-s + 2.53·35-s + 1.31·37-s + 1.28·39-s + 3.04·43-s + 0.875·47-s + 18/7·49-s − 0.412·53-s + 1.21·55-s − 0.529·57-s + 0.390·59-s − 1.28·61-s + 2.97·65-s − 1.22·67-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.003855279\)
\(L(\frac12)\) \(\approx\) \(6.003855279\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.470559617266477853109335520265, −9.240094318135156573014378604660, −9.107097894659908796841427844327, −8.741997778205243035007735862856, −8.226509065591357276840740260096, −7.902269515391642794068551855870, −7.43052843126784923830804002783, −7.12031664699832381613450815173, −6.28585863540952544590705266370, −5.98144365808840547702181627160, −5.73661244355422299089685552533, −5.47751162947620717730496558226, −4.48541590228767465398313202000, −4.31653107281507824602102119470, −3.85970190416332583000861366478, −3.30617585570210434707654021698, −2.31439329440424968443513837550, −2.15484216576729020701499530266, −1.37618930347234530013719263678, −1.21409277329025786161923642537, 1.21409277329025786161923642537, 1.37618930347234530013719263678, 2.15484216576729020701499530266, 2.31439329440424968443513837550, 3.30617585570210434707654021698, 3.85970190416332583000861366478, 4.31653107281507824602102119470, 4.48541590228767465398313202000, 5.47751162947620717730496558226, 5.73661244355422299089685552533, 5.98144365808840547702181627160, 6.28585863540952544590705266370, 7.12031664699832381613450815173, 7.43052843126784923830804002783, 7.902269515391642794068551855870, 8.226509065591357276840740260096, 8.741997778205243035007735862856, 9.107097894659908796841427844327, 9.240094318135156573014378604660, 9.470559617266477853109335520265

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