Properties

Label 4-2850e2-1.1-c1e2-0-19
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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 + 4·6-s + 4·7-s + 4·8-s + 3·9-s + 2·11-s + 6·12-s + 8·14-s + 5·16-s + 4·17-s + 6·18-s + 2·19-s + 8·21-s + 4·22-s − 2·23-s + 8·24-s + 4·27-s + 12·28-s + 6·29-s − 6·31-s + 6·32-s + 4·33-s + 8·34-s + 9·36-s + 4·37-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.51·7-s + 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 2.13·14-s + 5/4·16-s + 0.970·17-s + 1.41·18-s + 0.458·19-s + 1.74·21-s + 0.852·22-s − 0.417·23-s + 1.63·24-s + 0.769·27-s + 2.26·28-s + 1.11·29-s − 1.07·31-s + 1.06·32-s + 0.696·33-s + 1.37·34-s + 3/2·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(17.44359040\)
\(L(\frac12)\) \(\approx\) \(17.44359040\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 125 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 152 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 2 T + 113 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 203 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 204 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−8.737758727543438572287638875497, −8.653749543697073343755866390903, −7.911963501914263737797331977426, −7.88639056193247196846963749607, −7.41193256573456384568363244441, −7.34322251645105806821781545156, −6.53279609354272689207936999325, −6.32428577261191538069835441484, −5.81313881437308006709748361570, −5.40164789597058649026960225906, −4.92822969354287169868296173550, −4.68096876548028483421755905033, −4.08927976574825953133204088878, −4.01342753890289662619039195054, −3.29320357549393105643964805691, −3.06105256458612005816213438089, −2.44451831551702880233888861245, −2.02789944767745113755249956991, −1.40369334751528457070149741084, −1.11114235008707996922985062677, 1.11114235008707996922985062677, 1.40369334751528457070149741084, 2.02789944767745113755249956991, 2.44451831551702880233888861245, 3.06105256458612005816213438089, 3.29320357549393105643964805691, 4.01342753890289662619039195054, 4.08927976574825953133204088878, 4.68096876548028483421755905033, 4.92822969354287169868296173550, 5.40164789597058649026960225906, 5.81313881437308006709748361570, 6.32428577261191538069835441484, 6.53279609354272689207936999325, 7.34322251645105806821781545156, 7.41193256573456384568363244441, 7.88639056193247196846963749607, 7.911963501914263737797331977426, 8.653749543697073343755866390903, 8.737758727543438572287638875497

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