Properties

Label 4-9280e2-1.1-c1e2-0-7
Degree $4$
Conductor $86118400$
Sign $1$
Analytic cond. $5490.98$
Root an. cond. $8.60820$
Motivic weight $1$
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  + 3-s + 2·5-s + 7-s − 4·9-s + 2·11-s − 13-s + 2·15-s − 5·17-s − 10·19-s + 21-s + 5·23-s + 3·25-s − 6·27-s − 2·29-s − 31-s + 2·33-s + 2·35-s − 14·37-s − 39-s + 6·41-s + 13·43-s − 8·45-s − 16·47-s − 2·49-s − 5·51-s − 3·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s − 4/3·9-s + 0.603·11-s − 0.277·13-s + 0.516·15-s − 1.21·17-s − 2.29·19-s + 0.218·21-s + 1.04·23-s + 3/5·25-s − 1.15·27-s − 0.371·29-s − 0.179·31-s + 0.348·33-s + 0.338·35-s − 2.30·37-s − 0.160·39-s + 0.937·41-s + 1.98·43-s − 1.19·45-s − 2.33·47-s − 2/7·49-s − 0.700·51-s − 0.412·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86118400\)    =    \(2^{12} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5490.98\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 86118400,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 41 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 61 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 13 T + 127 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 21 T + 227 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 15 T + 171 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_4$ \( 1 + 17 T + 199 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 18 T + 202 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−7.55206064723423335593254502470, −6.99584813677151222459854240199, −6.89214496612914423220694980347, −6.57711820154874363071389545705, −6.15580567635365846634242216530, −5.89588607269668331921111960380, −5.47963124411511622957561202396, −5.20932837965744016296418115028, −4.63207304661744759982322548886, −4.57100548365004783540512273773, −3.94263195039716453567812476071, −3.63375901460991895247706476638, −3.22616781363895481552413773630, −2.58686133776375308009105927657, −2.40332193047745199439211773288, −2.16890695795740551330032630485, −1.59483722820858025670703466595, −1.15760134444192456447129299105, 0, 0, 1.15760134444192456447129299105, 1.59483722820858025670703466595, 2.16890695795740551330032630485, 2.40332193047745199439211773288, 2.58686133776375308009105927657, 3.22616781363895481552413773630, 3.63375901460991895247706476638, 3.94263195039716453567812476071, 4.57100548365004783540512273773, 4.63207304661744759982322548886, 5.20932837965744016296418115028, 5.47963124411511622957561202396, 5.89588607269668331921111960380, 6.15580567635365846634242216530, 6.57711820154874363071389545705, 6.89214496612914423220694980347, 6.99584813677151222459854240199, 7.55206064723423335593254502470

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