Properties

Label 4-2340e2-1.1-c1e2-0-3
Degree $4$
Conductor $5475600$
Sign $1$
Analytic cond. $349.129$
Root an. cond. $4.32261$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·13-s + 14·17-s − 10·23-s − 25-s + 20·29-s − 20·43-s − 11·49-s − 6·53-s − 6·61-s + 22·79-s + 20·101-s − 32·103-s − 6·107-s − 12·113-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.10·13-s + 3.39·17-s − 2.08·23-s − 1/5·25-s + 3.71·29-s − 3.04·43-s − 1.57·49-s − 0.824·53-s − 0.768·61-s + 2.47·79-s + 1.99·101-s − 3.15·103-s − 0.580·107-s − 1.12·113-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5475600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(349.129\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5475600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.089506923\)
\(L(\frac12)\) \(\approx\) \(2.089506923\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 129 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
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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

−9.432309939435422914674040208681, −8.492441588334282715469981505705, −8.354477193559563192615352072039, −7.923022058846467000296861489394, −7.909517179003454954694443801062, −7.37319795215400871023469472377, −6.79249690913896512162748077361, −6.33687979946903108045394213671, −6.26974604145328352693822555295, −5.56997482234549292974850643832, −5.20748452789313535586009164314, −4.82209976237409628280556190302, −4.56458745122432089651489905648, −3.77991099481154223672250760835, −3.36406648445537590876230411688, −3.04624712753436455867114710652, −2.56680637417157147439559997875, −1.73965112852503995105138626493, −1.33987262118126938726585967187, −0.51610018215352404524271297822, 0.51610018215352404524271297822, 1.33987262118126938726585967187, 1.73965112852503995105138626493, 2.56680637417157147439559997875, 3.04624712753436455867114710652, 3.36406648445537590876230411688, 3.77991099481154223672250760835, 4.56458745122432089651489905648, 4.82209976237409628280556190302, 5.20748452789313535586009164314, 5.56997482234549292974850643832, 6.26974604145328352693822555295, 6.33687979946903108045394213671, 6.79249690913896512162748077361, 7.37319795215400871023469472377, 7.909517179003454954694443801062, 7.923022058846467000296861489394, 8.354477193559563192615352072039, 8.492441588334282715469981505705, 9.432309939435422914674040208681

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