Properties

Label 4-901404-1.1-c1e2-0-3
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  − 4-s − 7-s − 3·9-s + 6·13-s + 16-s + 8·25-s + 28-s + 6·31-s + 3·36-s + 8·37-s + 8·43-s + 49-s − 6·52-s + 12·61-s + 3·63-s − 64-s + 4·67-s − 17·73-s − 2·79-s + 9·81-s − 6·91-s − 8·100-s + 18·103-s − 28·109-s − 112-s − 18·117-s + 4·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.377·7-s − 9-s + 1.66·13-s + 1/4·16-s + 8/5·25-s + 0.188·28-s + 1.07·31-s + 1/2·36-s + 1.31·37-s + 1.21·43-s + 1/7·49-s − 0.832·52-s + 1.53·61-s + 0.377·63-s − 1/8·64-s + 0.488·67-s − 1.98·73-s − 0.225·79-s + 81-s − 0.628·91-s − 4/5·100-s + 1.77·103-s − 2.68·109-s − 0.0944·112-s − 1.66·117-s + 4/11·121-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

\(L(1)\) \(\approx\) \(1.746999853\)
\(L(\frac12)\) \(\approx\) \(1.746999853\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( 1 + T \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 16 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.160832406868106744472454397254, −7.980540082647813317765940563750, −7.29295210132032211003099308997, −6.75572945672478158650638296893, −6.30569460926660048049739706937, −5.99444531876732676687618169733, −5.55668669444676963497438290654, −5.02203868791154749090748662406, −4.45541972491189446182909000979, −3.96490027234389551938265787802, −3.46865143626618368150277943152, −2.83981981195085262248216410009, −2.50518462051537220186802360737, −1.30280236135806803400948722458, −0.70666920691753134427065041706, 0.70666920691753134427065041706, 1.30280236135806803400948722458, 2.50518462051537220186802360737, 2.83981981195085262248216410009, 3.46865143626618368150277943152, 3.96490027234389551938265787802, 4.45541972491189446182909000979, 5.02203868791154749090748662406, 5.55668669444676963497438290654, 5.99444531876732676687618169733, 6.30569460926660048049739706937, 6.75572945672478158650638296893, 7.29295210132032211003099308997, 7.980540082647813317765940563750, 8.160832406868106744472454397254

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