Properties

Label 4-3920e2-1.1-c1e2-0-2
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
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  + 4·3-s − 2·5-s + 8·9-s − 4·11-s + 4·13-s − 8·15-s − 8·17-s + 4·19-s + 8·23-s + 3·25-s + 12·27-s − 4·29-s − 16·33-s − 4·37-s + 16·39-s − 8·41-s + 12·43-s − 16·45-s + 16·47-s − 32·51-s − 4·53-s + 8·55-s + 16·57-s + 20·59-s + 4·61-s − 8·65-s + 8·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 8/3·9-s − 1.20·11-s + 1.10·13-s − 2.06·15-s − 1.94·17-s + 0.917·19-s + 1.66·23-s + 3/5·25-s + 2.30·27-s − 0.742·29-s − 2.78·33-s − 0.657·37-s + 2.56·39-s − 1.24·41-s + 1.82·43-s − 2.38·45-s + 2.33·47-s − 4.48·51-s − 0.549·53-s + 1.07·55-s + 2.11·57-s + 2.60·59-s + 0.512·61-s − 0.992·65-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.380432687\)
\(L(\frac12)\) \(\approx\) \(5.380432687\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 20 T + 216 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 152 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−8.510681357508977944207093342558, −8.498721180093561669510885365990, −7.907096097715219973485383741831, −7.78553553643322768018561132562, −7.20501363643117499511727013126, −7.15178722084264799934689833095, −6.60315553458388563385291702998, −6.31317563288933886098026474916, −5.39500350223783840427539990003, −5.24575071850530207553389623313, −4.90164310537265708096105500017, −4.07831984226271294562981588852, −3.91701709551788122537831939669, −3.72188982284049364514294833987, −3.01900199660792516533741699049, −2.85082410284428121753506914211, −2.30387834697612667523824269756, −2.13805393770224068609389708092, −1.19358818603872925223689358355, −0.60511987838352224882598091851, 0.60511987838352224882598091851, 1.19358818603872925223689358355, 2.13805393770224068609389708092, 2.30387834697612667523824269756, 2.85082410284428121753506914211, 3.01900199660792516533741699049, 3.72188982284049364514294833987, 3.91701709551788122537831939669, 4.07831984226271294562981588852, 4.90164310537265708096105500017, 5.24575071850530207553389623313, 5.39500350223783840427539990003, 6.31317563288933886098026474916, 6.60315553458388563385291702998, 7.15178722084264799934689833095, 7.20501363643117499511727013126, 7.78553553643322768018561132562, 7.907096097715219973485383741831, 8.498721180093561669510885365990, 8.510681357508977944207093342558

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