Properties

Label 4-901404-1.1-c1e2-0-6
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 7-s − 2·9-s − 12-s − 3·13-s + 16-s − 6·19-s − 21-s + 7·25-s + 5·27-s + 28-s − 31-s − 2·36-s + 3·37-s + 3·39-s − 8·43-s − 48-s + 49-s − 3·52-s + 6·57-s + 15·61-s − 2·63-s + 64-s − 67-s − 3·73-s − 7·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 0.377·7-s − 2/3·9-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 1.37·19-s − 0.218·21-s + 7/5·25-s + 0.962·27-s + 0.188·28-s − 0.179·31-s − 1/3·36-s + 0.493·37-s + 0.480·39-s − 1.21·43-s − 0.144·48-s + 1/7·49-s − 0.416·52-s + 0.794·57-s + 1.92·61-s − 0.251·63-s + 1/8·64-s − 0.122·67-s − 0.351·73-s − 0.808·75-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: \(1\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + T + p T^{2} \)
7$C_1$ \( 1 - T \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \)
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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

−8.078382205826496916400872836170, −7.39055415768506913216291331873, −7.04124135696293323642508536908, −6.53166916779015515552472946730, −6.32685964902407679111258013835, −5.67379505479463132593870072494, −5.24507015885076221783331950242, −4.86397666839575488824992131573, −4.37308609247023368982511423061, −3.74328371692022032483950173050, −3.01249087105239852332863565166, −2.54577378313166876214006499087, −2.00434696354280387396991128910, −1.08521721399422424433640799951, 0, 1.08521721399422424433640799951, 2.00434696354280387396991128910, 2.54577378313166876214006499087, 3.01249087105239852332863565166, 3.74328371692022032483950173050, 4.37308609247023368982511423061, 4.86397666839575488824992131573, 5.24507015885076221783331950242, 5.67379505479463132593870072494, 6.32685964902407679111258013835, 6.53166916779015515552472946730, 7.04124135696293323642508536908, 7.39055415768506913216291331873, 8.078382205826496916400872836170

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