Properties

Label 4-901404-1.1-c1e2-0-2
Degree $4$
Conductor $901404$
Sign $1$
Analytic cond. $57.4743$
Root an. cond. $2.75339$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4-s − 7-s + 6·9-s + 3·12-s + 3·13-s + 16-s + 3·21-s − 25-s − 9·27-s + 28-s + 9·31-s − 6·36-s + 5·37-s − 9·39-s + 8·43-s − 3·48-s + 49-s − 3·52-s + 9·61-s − 6·63-s − 64-s − 5·67-s + 73-s + 3·75-s + 4·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s − 1/2·4-s − 0.377·7-s + 2·9-s + 0.866·12-s + 0.832·13-s + 1/4·16-s + 0.654·21-s − 1/5·25-s − 1.73·27-s + 0.188·28-s + 1.61·31-s − 36-s + 0.821·37-s − 1.44·39-s + 1.21·43-s − 0.433·48-s + 1/7·49-s − 0.416·52-s + 1.15·61-s − 0.755·63-s − 1/8·64-s − 0.610·67-s + 0.117·73-s + 0.346·75-s + 0.450·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(901404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73\)
Sign: $1$
Analytic conductor: \(57.4743\)
Root analytic conductor: \(2.75339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{901404} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9117869335\)
\(L(\frac12)\) \(\approx\) \(0.9117869335\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
7$C_1$ \( 1 + T \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−8.318008438991289993315007888666, −7.48245353184456152941549350213, −7.35156564777050085522577847765, −6.60638182693326161618251056166, −6.31635538667819651049251483209, −5.94468201904435523015631772120, −5.61239355675081315079617672570, −5.01077757322795603322785085211, −4.56134135438064287025250025384, −4.19938866623469113746777941571, −3.63784553140918674542066302688, −2.95429116618460432717681410796, −2.11105866831813092685590283972, −1.12106303563957554716196069464, −0.61577346227566000338851588569, 0.61577346227566000338851588569, 1.12106303563957554716196069464, 2.11105866831813092685590283972, 2.95429116618460432717681410796, 3.63784553140918674542066302688, 4.19938866623469113746777941571, 4.56134135438064287025250025384, 5.01077757322795603322785085211, 5.61239355675081315079617672570, 5.94468201904435523015631772120, 6.31635538667819651049251483209, 6.60638182693326161618251056166, 7.35156564777050085522577847765, 7.48245353184456152941549350213, 8.318008438991289993315007888666

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