Properties

Label 4-8040e2-1.1-c1e2-0-0
Degree $4$
Conductor $64641600$
Sign $1$
Analytic cond. $4121.60$
Root an. cond. $8.01247$
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 + 7-s + 3·9-s + 7·11-s + 2·13-s + 4·15-s − 6·17-s + 6·19-s + 2·21-s − 8·23-s + 3·25-s + 4·27-s + 6·29-s + 6·31-s + 14·33-s + 2·35-s + 37-s + 4·39-s + 12·41-s + 14·43-s + 6·45-s − 2·47-s − 9·49-s − 12·51-s − 2·53-s + 14·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.377·7-s + 9-s + 2.11·11-s + 0.554·13-s + 1.03·15-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 1.66·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 1.07·31-s + 2.43·33-s + 0.338·35-s + 0.164·37-s + 0.640·39-s + 1.87·41-s + 2.13·43-s + 0.894·45-s − 0.291·47-s − 9/7·49-s − 1.68·51-s − 0.274·53-s + 1.88·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64641600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64641600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64641600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 67^{2}\)
Sign: $1$
Analytic conductor: \(4121.60\)
Root analytic conductor: \(8.01247\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 64641600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(10.86725402\)
\(L(\frac12)\) \(\approx\) \(10.86725402\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
67$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 19 T + 228 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_4$ \( 1 + 23 T + 294 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 25 T + 346 T^{2} - 25 p T^{3} + p^{2} T^{4} \)
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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

−8.101997652440054771685517138884, −7.60458727201949444606761689478, −7.27809998907248484671758255193, −7.19424823207499878875399265354, −6.33997071603033240948723384780, −6.33195227484538973456764689683, −6.06079224872589101788506472357, −5.92492918633722996570413706114, −4.88933334625498304352646673370, −4.87544408724898724780261989330, −4.29508896480258978866115539792, −4.21575450108730520272455516370, −3.51399730186506685658488625481, −3.51183954763214654532974670335, −2.62996698887298128459456258773, −2.60818400288396391195023370218, −1.91597852867178157556792825208, −1.69509706841606224362470294959, −1.06311274097880722048902186751, −0.807333485565959124916884558276, 0.807333485565959124916884558276, 1.06311274097880722048902186751, 1.69509706841606224362470294959, 1.91597852867178157556792825208, 2.60818400288396391195023370218, 2.62996698887298128459456258773, 3.51183954763214654532974670335, 3.51399730186506685658488625481, 4.21575450108730520272455516370, 4.29508896480258978866115539792, 4.87544408724898724780261989330, 4.88933334625498304352646673370, 5.92492918633722996570413706114, 6.06079224872589101788506472357, 6.33195227484538973456764689683, 6.33997071603033240948723384780, 7.19424823207499878875399265354, 7.27809998907248484671758255193, 7.60458727201949444606761689478, 8.101997652440054771685517138884

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