Properties

Label 4-96e4-1.1-c1e2-0-9
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  + 8·11-s − 12·17-s − 4·19-s − 2·25-s + 20·41-s + 12·43-s + 4·49-s − 8·67-s + 32·73-s + 32·83-s − 28·89-s − 8·97-s + 8·107-s + 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.41·11-s − 2.91·17-s − 0.917·19-s − 2/5·25-s + 3.12·41-s + 1.82·43-s + 4/7·49-s − 0.977·67-s + 3.74·73-s + 3.51·83-s − 2.96·89-s − 0.812·97-s + 0.773·107-s + 1.12·113-s + 2.36·121-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 − 0.615·169-s + 0.0760·173-s + 0.0747·179-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

\(L(1)\) \(\approx\) \(3.868780646\)
\(L(\frac12)\) \(\approx\) \(3.868780646\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−7.87781076718369081753609279898, −7.46711368586190404627363655533, −7.11497857117867050492684260229, −6.81426977650685007789003506336, −6.42571234010796720616788010946, −6.35792083247936148795549021974, −5.93586110990468018698790064105, −5.70583441821828038836010520928, −4.98263281409415164383855533722, −4.63055244073769142094778628754, −4.21048939951121476702749181852, −4.12863762399576450367953546486, −3.88942158791440307860429849257, −3.38112227141562985392063126173, −2.54537806902042514566089254365, −2.47457607562477944304603797251, −1.96995799776714901236618145174, −1.62067131650310832242431645548, −0.76081128402105898024195550019, −0.60954835033349355277274915156, 0.60954835033349355277274915156, 0.76081128402105898024195550019, 1.62067131650310832242431645548, 1.96995799776714901236618145174, 2.47457607562477944304603797251, 2.54537806902042514566089254365, 3.38112227141562985392063126173, 3.88942158791440307860429849257, 4.12863762399576450367953546486, 4.21048939951121476702749181852, 4.63055244073769142094778628754, 4.98263281409415164383855533722, 5.70583441821828038836010520928, 5.93586110990468018698790064105, 6.35792083247936148795549021974, 6.42571234010796720616788010946, 6.81426977650685007789003506336, 7.11497857117867050492684260229, 7.46711368586190404627363655533, 7.87781076718369081753609279898

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