Properties

Label 4-2240e2-1.1-c1e2-0-16
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 2·7-s − 9-s − 11-s − 3·13-s + 2·15-s + 5·17-s + 2·21-s + 3·25-s − 29-s + 2·31-s + 33-s + 4·35-s − 6·37-s + 3·39-s + 4·41-s − 8·43-s + 2·45-s − 11·47-s + 3·49-s − 5·51-s − 18·53-s + 2·55-s + 4·59-s + 4·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.755·7-s − 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.516·15-s + 1.21·17-s + 0.436·21-s + 3/5·25-s − 0.185·29-s + 0.359·31-s + 0.174·33-s + 0.676·35-s − 0.986·37-s + 0.480·39-s + 0.624·41-s − 1.21·43-s + 0.298·45-s − 1.60·47-s + 3/7·49-s − 0.700·51-s − 2.47·53-s + 0.269·55-s + 0.520·59-s + 0.512·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 3 T - 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−8.739258981859685727783109207165, −8.465171310912883291222650311507, −7.86218332116970317587974714569, −7.79767775479679517618969943550, −7.38475579020328726436737389070, −6.82595158847369238004008889972, −6.39141597342121333975559988429, −6.35874780034238525423776996856, −5.46572023579994268774300016196, −5.33663300685982271466866270898, −4.97306225676570154044469298428, −4.44152135326070623920419020624, −3.82695145689767356459927985173, −3.54418853156303576582044155516, −2.86009336980223243862642897105, −2.81681419792061339744465219807, −1.79870447151663116465203390608, −1.16319875709089676757108793083, 0, 0, 1.16319875709089676757108793083, 1.79870447151663116465203390608, 2.81681419792061339744465219807, 2.86009336980223243862642897105, 3.54418853156303576582044155516, 3.82695145689767356459927985173, 4.44152135326070623920419020624, 4.97306225676570154044469298428, 5.33663300685982271466866270898, 5.46572023579994268774300016196, 6.35874780034238525423776996856, 6.39141597342121333975559988429, 6.82595158847369238004008889972, 7.38475579020328726436737389070, 7.79767775479679517618969943550, 7.86218332116970317587974714569, 8.465171310912883291222650311507, 8.739258981859685727783109207165

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