Properties

Label 4-3200e2-1.1-c1e2-0-22
Degree $4$
Conductor $10240000$
Sign $1$
Analytic cond. $652.911$
Root an. cond. $5.05491$
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 + 4·7-s − 9-s + 2·11-s + 6·17-s + 6·19-s − 8·21-s + 4·23-s + 6·27-s − 16·29-s + 4·31-s − 4·33-s − 4·37-s + 10·41-s − 20·43-s + 8·47-s + 6·49-s − 12·51-s + 4·53-s − 12·57-s + 4·59-s − 12·61-s − 4·63-s + 14·67-s − 8·69-s − 8·71-s − 6·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s − 1/3·9-s + 0.603·11-s + 1.45·17-s + 1.37·19-s − 1.74·21-s + 0.834·23-s + 1.15·27-s − 2.97·29-s + 0.718·31-s − 0.696·33-s − 0.657·37-s + 1.56·41-s − 3.04·43-s + 1.16·47-s + 6/7·49-s − 1.68·51-s + 0.549·53-s − 1.58·57-s + 0.520·59-s − 1.53·61-s − 0.503·63-s + 1.71·67-s − 0.963·69-s − 0.949·71-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10240000\)    =    \(2^{14} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(652.911\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.245123870\)
\(L(\frac12)\) \(\approx\) \(2.245123870\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 147 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 20 T + 250 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 157 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 251 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.004226813107531565084218041729, −8.387793127411064329101240827808, −7.999074365969824494290242841104, −7.59489059071052323706659983290, −7.56887103064768595205763100330, −7.03676014516620699102890183706, −6.41030030655787428061247477010, −6.24994609108507013909849993427, −5.60792138824045654886823473972, −5.35221018591912918383327415071, −5.15976134122521946423128357563, −4.97880710637258612649134856336, −4.17038576256479047738542312157, −3.85383871969369583013552300199, −3.15978815668361773663467847378, −3.10073598257651528713765136961, −1.96881282638433772598652182865, −1.80009446496141328112468694990, −1.02720602017096475216196914316, −0.60600542720599434226467620275, 0.60600542720599434226467620275, 1.02720602017096475216196914316, 1.80009446496141328112468694990, 1.96881282638433772598652182865, 3.10073598257651528713765136961, 3.15978815668361773663467847378, 3.85383871969369583013552300199, 4.17038576256479047738542312157, 4.97880710637258612649134856336, 5.15976134122521946423128357563, 5.35221018591912918383327415071, 5.60792138824045654886823473972, 6.24994609108507013909849993427, 6.41030030655787428061247477010, 7.03676014516620699102890183706, 7.56887103064768595205763100330, 7.59489059071052323706659983290, 7.999074365969824494290242841104, 8.387793127411064329101240827808, 9.004226813107531565084218041729

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