Properties

Label 4-1200e2-1.1-c3e2-0-30
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $5012.96$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 6·7-s + 27·9-s + 42·11-s − 78·13-s − 102·17-s − 56·19-s + 36·21-s − 48·23-s + 108·27-s − 318·29-s − 52·31-s + 252·33-s − 306·37-s − 468·39-s − 408·41-s + 120·43-s + 180·47-s − 290·49-s − 612·51-s − 402·53-s − 336·57-s + 186·59-s + 340·61-s + 162·63-s + 732·67-s − 288·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.323·7-s + 9-s + 1.15·11-s − 1.66·13-s − 1.45·17-s − 0.676·19-s + 0.374·21-s − 0.435·23-s + 0.769·27-s − 2.03·29-s − 0.301·31-s + 1.32·33-s − 1.35·37-s − 1.92·39-s − 1.55·41-s + 0.425·43-s + 0.558·47-s − 0.845·49-s − 1.68·51-s − 1.04·53-s − 0.780·57-s + 0.410·59-s + 0.713·61-s + 0.323·63-s + 1.33·67-s − 0.502·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5012.96\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1440000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
5 \( 1 \)
good7$D_{4}$ \( 1 - 6 T + 326 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 42 T + 2734 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 6 p T + 5546 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 6 p T + 11402 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 56 T + 8598 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 48 T + 24254 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 52 T + 58782 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 306 T + 94826 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 120 T + 68150 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 180 T + 128990 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 402 T + 269234 T^{2} + 402 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 732 T + 698582 T^{2} - 732 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1332 T + 1102034 T^{2} + 1332 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 380 T + 99678 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 984 T + 942182 T^{2} + 984 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1116 T + 1508758 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 768 T + 1382402 T^{2} - 768 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.044398746382054313670974091992, −8.932228356681616004460172006328, −8.325205670106297362783839144532, −8.073413254060382002252391792765, −7.54485599088918513206111052289, −7.07663249571948796527320413882, −6.74620077163965259877013419186, −6.60614180145211059896373100983, −5.63634927037724485299927798248, −5.38574824784766810406018975284, −4.61724493613106626815371744599, −4.43915116993803682161255823130, −3.74636879365726534589709906161, −3.61413686011451003933891708712, −2.68352074430360340472209109722, −2.35605267086697107036003493447, −1.72066153632002513608749342338, −1.50229983338452310594248340169, 0, 0, 1.50229983338452310594248340169, 1.72066153632002513608749342338, 2.35605267086697107036003493447, 2.68352074430360340472209109722, 3.61413686011451003933891708712, 3.74636879365726534589709906161, 4.43915116993803682161255823130, 4.61724493613106626815371744599, 5.38574824784766810406018975284, 5.63634927037724485299927798248, 6.60614180145211059896373100983, 6.74620077163965259877013419186, 7.07663249571948796527320413882, 7.54485599088918513206111052289, 8.073413254060382002252391792765, 8.325205670106297362783839144532, 8.932228356681616004460172006328, 9.044398746382054313670974091992

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