Properties

Label 4-2070e2-1.1-c1e2-0-1
Degree $4$
Conductor $4284900$
Sign $1$
Analytic cond. $273.208$
Root an. cond. $4.06559$
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  − 2·2-s + 3·4-s − 2·5-s + 7-s − 4·8-s + 4·10-s − 11-s − 3·13-s − 2·14-s + 5·16-s − 17-s − 3·19-s − 6·20-s + 2·22-s − 2·23-s + 3·25-s + 6·26-s + 3·28-s + 14·29-s + 7·31-s − 6·32-s + 2·34-s − 2·35-s + 4·37-s + 6·38-s + 8·40-s + 9·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s − 1.41·8-s + 1.26·10-s − 0.301·11-s − 0.832·13-s − 0.534·14-s + 5/4·16-s − 0.242·17-s − 0.688·19-s − 1.34·20-s + 0.426·22-s − 0.417·23-s + 3/5·25-s + 1.17·26-s + 0.566·28-s + 2.59·29-s + 1.25·31-s − 1.06·32-s + 0.342·34-s − 0.338·35-s + 0.657·37-s + 0.973·38-s + 1.26·40-s + 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4284900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(273.208\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4284900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045756397\)
\(L(\frac12)\) \(\approx\) \(1.045756397\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 9 T + p T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
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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

−9.223470128264673063072497875000, −8.915919089267394368139024097000, −8.326617505517797241380144977711, −8.193832656684335668977110465955, −7.919274553111697184622416661461, −7.67707462950570956460978681126, −6.95851054108973961374312161214, −6.74188919941959306768373955162, −6.41471522684785427656491994896, −6.00760005514998738072551903495, −5.22464210615257006858018019389, −4.92810056619173249586244672859, −4.34509925821010978980810211039, −4.15340517730267265336944226327, −3.15578636193988013655177999170, −3.01374878089399841621000142033, −2.20303926815927969478566357017, −2.05326656139358273818644960548, −0.803817388064460988594638353418, −0.67445417495635941704012673035, 0.67445417495635941704012673035, 0.803817388064460988594638353418, 2.05326656139358273818644960548, 2.20303926815927969478566357017, 3.01374878089399841621000142033, 3.15578636193988013655177999170, 4.15340517730267265336944226327, 4.34509925821010978980810211039, 4.92810056619173249586244672859, 5.22464210615257006858018019389, 6.00760005514998738072551903495, 6.41471522684785427656491994896, 6.74188919941959306768373955162, 6.95851054108973961374312161214, 7.67707462950570956460978681126, 7.919274553111697184622416661461, 8.193832656684335668977110465955, 8.326617505517797241380144977711, 8.915919089267394368139024097000, 9.223470128264673063072497875000

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