Properties

Label 4-5520e2-1.1-c1e2-0-3
Degree $4$
Conductor $30470400$
Sign $1$
Analytic cond. $1942.81$
Root an. cond. $6.63908$
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  + 2·3-s − 2·5-s + 2·7-s + 3·9-s + 8·11-s + 4·13-s − 4·15-s + 2·17-s + 4·19-s + 4·21-s + 2·23-s + 3·25-s + 4·27-s − 10·29-s + 14·31-s + 16·33-s − 4·35-s + 6·37-s + 8·39-s − 18·41-s − 12·43-s − 6·45-s + 20·47-s − 3·49-s + 4·51-s + 6·53-s − 16·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 2.41·11-s + 1.10·13-s − 1.03·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s − 1.85·29-s + 2.51·31-s + 2.78·33-s − 0.676·35-s + 0.986·37-s + 1.28·39-s − 2.81·41-s − 1.82·43-s − 0.894·45-s + 2.91·47-s − 3/7·49-s + 0.560·51-s + 0.824·53-s − 2.15·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30470400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1942.81\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 30470400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.647805143\)
\(L(\frac12)\) \(\approx\) \(7.647805143\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 8 T + 36 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 14 T + 103 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 18 T + 161 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 20 T + 192 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 97 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 136 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 164 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T - 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.364869358971764432486544232724, −8.186838891654689193585792915699, −7.61038407133851102709161456071, −7.33564887958641428986346291676, −6.89762682289004846920100763336, −6.82265157402010568278774762325, −6.19398484201687217814882751083, −6.01062516328945728729552727363, −5.21620180852674665440509122172, −5.15100354273576273189962222241, −4.32497321234493354863663967334, −4.25640189290561918587938977529, −3.79033189161617006723110801665, −3.63035693658830653858738988437, −3.00821290859783217813905802207, −2.90925675270995681314017191979, −1.90587536437901508729160348106, −1.64256685714035588528212738351, −1.13493705019001449289508767985, −0.78400020226473795810350419120, 0.78400020226473795810350419120, 1.13493705019001449289508767985, 1.64256685714035588528212738351, 1.90587536437901508729160348106, 2.90925675270995681314017191979, 3.00821290859783217813905802207, 3.63035693658830653858738988437, 3.79033189161617006723110801665, 4.25640189290561918587938977529, 4.32497321234493354863663967334, 5.15100354273576273189962222241, 5.21620180852674665440509122172, 6.01062516328945728729552727363, 6.19398484201687217814882751083, 6.82265157402010568278774762325, 6.89762682289004846920100763336, 7.33564887958641428986346291676, 7.61038407133851102709161456071, 8.186838891654689193585792915699, 8.364869358971764432486544232724

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