Properties

Label 4-5610e2-1.1-c1e2-0-0
Degree $4$
Conductor $31472100$
Sign $1$
Analytic cond. $2006.68$
Root an. cond. $6.69298$
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  − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s − 4·10-s + 2·11-s − 6·12-s − 8·13-s + 2·14-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s + 8·19-s + 6·20-s + 2·21-s − 4·22-s − 9·23-s + 8·24-s + 3·25-s + 16·26-s − 4·27-s − 3·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.603·11-s − 1.73·12-s − 2.21·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 1.83·19-s + 1.34·20-s + 0.436·21-s − 0.852·22-s − 1.87·23-s + 1.63·24-s + 3/5·25-s + 3.13·26-s − 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31472100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31472100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2006.68\)
Root analytic conductor: \(6.69298\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 31472100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9049055749\)
\(L(\frac12)\) \(\approx\) \(0.9049055749\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 62 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 15 T + 134 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 154 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 7 p T^{3} + p^{2} T^{4} \)
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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

−8.058304830513310511418852477823, −8.044336531667542665213937731954, −7.41478031892098598841737577565, −7.35483143834538702932315332468, −7.02548105609422570947197930081, −6.62797740994964576833693780278, −6.20850301370970285413606459607, −5.79943890746369625948709625261, −5.51266963798826842399347047975, −5.46658654340193907174344207488, −4.63164546312937028143441318362, −4.58430035306683513395586373239, −3.69746589021308049170562338408, −3.49237602871960727439293704054, −2.72374669469739254582712587344, −2.45410394658772014578618962496, −1.73917000213990662085948766465, −1.71493909730738724934212066852, −0.71719443631182514220201277305, −0.49688705346073905235472774821, 0.49688705346073905235472774821, 0.71719443631182514220201277305, 1.71493909730738724934212066852, 1.73917000213990662085948766465, 2.45410394658772014578618962496, 2.72374669469739254582712587344, 3.49237602871960727439293704054, 3.69746589021308049170562338408, 4.58430035306683513395586373239, 4.63164546312937028143441318362, 5.46658654340193907174344207488, 5.51266963798826842399347047975, 5.79943890746369625948709625261, 6.20850301370970285413606459607, 6.62797740994964576833693780278, 7.02548105609422570947197930081, 7.35483143834538702932315332468, 7.41478031892098598841737577565, 8.044336531667542665213937731954, 8.058304830513310511418852477823

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