Properties

Label 4-96e4-1.1-c1e2-0-4
Degree $4$
Conductor $84934656$
Sign $1$
Analytic cond. $5415.50$
Root an. cond. $8.57846$
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  − 4·7-s − 4·17-s + 12·23-s − 8·25-s − 16·31-s + 16·47-s − 2·49-s + 20·71-s + 8·73-s − 8·89-s − 4·97-s + 12·103-s − 12·113-s + 16·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s − 24·169-s + 173-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.970·17-s + 2.50·23-s − 8/5·25-s − 2.87·31-s + 2.33·47-s − 2/7·49-s + 2.37·71-s + 0.936·73-s − 0.847·89-s − 0.406·97-s + 1.18·103-s − 1.12·113-s + 1.46·119-s − 1.81·121-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 − 3.78·161-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5415.50\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84934656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744796432\)
\(L(\frac12)\) \(\approx\) \(1.744796432\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 164 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−7.63438919338293602430347308191, −7.52744505199193941504454716700, −7.20225080761058324462132967572, −6.81661444948021737503267202348, −6.56198873272250189527218510595, −6.33867457918694897919168326919, −5.75223234913966853225051170812, −5.55508660314495972686547193038, −5.16703740086058051182335791033, −4.92691908064845991080776551356, −4.15550209911613096501062135269, −4.06268716982881348786700568684, −3.55309748625572094089636450214, −3.38240327711582271964943877296, −2.70050193025494423084594068380, −2.65307163020726127418128630730, −1.78506719443029873857560523738, −1.76897001394204915176789813349, −0.68930109206845918516204097731, −0.42721575514083964807241665263, 0.42721575514083964807241665263, 0.68930109206845918516204097731, 1.76897001394204915176789813349, 1.78506719443029873857560523738, 2.65307163020726127418128630730, 2.70050193025494423084594068380, 3.38240327711582271964943877296, 3.55309748625572094089636450214, 4.06268716982881348786700568684, 4.15550209911613096501062135269, 4.92691908064845991080776551356, 5.16703740086058051182335791033, 5.55508660314495972686547193038, 5.75223234913966853225051170812, 6.33867457918694897919168326919, 6.56198873272250189527218510595, 6.81661444948021737503267202348, 7.20225080761058324462132967572, 7.52744505199193941504454716700, 7.63438919338293602430347308191

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