Properties

Label 4-1470e2-1.1-c1e2-0-12
Degree $4$
Conductor $2160900$
Sign $1$
Analytic cond. $137.780$
Root an. cond. $3.42607$
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-s + 3-s − 5-s − 6-s + 8-s + 10-s − 4·11-s − 4·13-s − 15-s − 16-s − 2·17-s + 4·19-s + 4·22-s + 8·23-s + 24-s + 4·26-s − 27-s + 12·29-s + 30-s + 8·31-s − 4·33-s + 2·34-s + 2·37-s − 4·38-s − 4·39-s − 40-s + 4·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.10·13-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.852·22-s + 1.66·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 2.22·29-s + 0.182·30-s + 1.43·31-s − 0.696·33-s + 0.342·34-s + 0.328·37-s − 0.648·38-s − 0.640·39-s − 0.158·40-s + 0.624·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(137.780\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.205833642\)
\(L(\frac12)\) \(\approx\) \(1.205833642\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−9.624555245572691207547369531836, −9.367917707914361146225118573668, −8.771488972154385057655487978209, −8.545994369597774701034142300739, −8.106759564129112732155841772358, −7.78309140646253881266458062402, −7.55516461665085885235739113654, −6.98844568775064610371432386998, −6.47850207074394746548164556853, −6.39195419311599498297393451808, −5.19615608224426043602634059511, −5.17279028966263741706147033597, −4.73839018213162375176045640831, −4.37733798685066541506852847064, −3.32422285334760083178589690144, −3.24354806167984956244132997803, −2.56115017801408889501986885816, −2.24827803507575228709301800556, −1.19622712247552016307997896447, −0.53129358613126882268842931066, 0.53129358613126882268842931066, 1.19622712247552016307997896447, 2.24827803507575228709301800556, 2.56115017801408889501986885816, 3.24354806167984956244132997803, 3.32422285334760083178589690144, 4.37733798685066541506852847064, 4.73839018213162375176045640831, 5.17279028966263741706147033597, 5.19615608224426043602634059511, 6.39195419311599498297393451808, 6.47850207074394746548164556853, 6.98844568775064610371432386998, 7.55516461665085885235739113654, 7.78309140646253881266458062402, 8.106759564129112732155841772358, 8.545994369597774701034142300739, 8.771488972154385057655487978209, 9.367917707914361146225118573668, 9.624555245572691207547369531836

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