Properties

Label 4-70e4-1.1-c1e2-0-14
Degree $4$
Conductor $24010000$
Sign $1$
Analytic cond. $1530.89$
Root an. cond. $6.25513$
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  + 2·9-s + 8·11-s + 8·23-s + 16·29-s + 16·37-s + 8·43-s − 20·53-s + 16·79-s − 5·81-s + 16·99-s − 16·107-s − 16·109-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 + 181-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.41·11-s + 1.66·23-s + 2.97·29-s + 2.63·37-s + 1.21·43-s − 2.74·53-s + 1.80·79-s − 5/9·81-s + 1.60·99-s − 1.54·107-s − 1.53·109-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 + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1530.89\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24010000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.431489508\)
\(L(\frac12)\) \(\approx\) \(5.431489508\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 192 T^{2} + p^{2} T^{4} \)
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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

−8.261230630956351170838363690348, −8.254827282833983292048350942376, −7.71045004352119243779776988851, −7.30331539016476462806141114305, −6.93481390856791769066453092985, −6.45831871903475564891624811743, −6.41393421431243039060607194330, −6.22615383318520579364249974873, −5.51997141698656966784947398309, −5.06056133590690184033807746277, −4.50796088206760797707562016831, −4.42885631005143267058009024336, −4.12581084462365198197487946089, −3.55209071954943527049885074619, −2.93718357607812468484050016290, −2.88028168055179038414671515265, −2.12964667996718740327044579309, −1.41474289686561157280179280886, −1.07768887130158783799120547121, −0.798854503574070641871186099960, 0.798854503574070641871186099960, 1.07768887130158783799120547121, 1.41474289686561157280179280886, 2.12964667996718740327044579309, 2.88028168055179038414671515265, 2.93718357607812468484050016290, 3.55209071954943527049885074619, 4.12581084462365198197487946089, 4.42885631005143267058009024336, 4.50796088206760797707562016831, 5.06056133590690184033807746277, 5.51997141698656966784947398309, 6.22615383318520579364249974873, 6.41393421431243039060607194330, 6.45831871903475564891624811743, 6.93481390856791769066453092985, 7.30331539016476462806141114305, 7.71045004352119243779776988851, 8.254827282833983292048350942376, 8.261230630956351170838363690348

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