Properties

Label 4-111132-1.1-c1e2-0-0
Degree $4$
Conductor $111132$
Sign $1$
Analytic cond. $7.08587$
Root an. cond. $1.63154$
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 + 16-s − 6·25-s − 28-s + 20·37-s + 8·43-s + 49-s − 64-s + 8·67-s + 6·100-s + 4·109-s + 112-s + 6·121-s + 127-s + 131-s + 137-s + 139-s − 20·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·172-s + 173-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.377·7-s + 1/4·16-s − 6/5·25-s − 0.188·28-s + 3.28·37-s + 1.21·43-s + 1/7·49-s − 1/8·64-s + 0.977·67-s + 3/5·100-s + 0.383·109-s + 0.0944·112-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.609·172-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(111132\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(7.08587\)
Root analytic conductor: \(1.63154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 111132,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.381015326\)
\(L(\frac12)\) \(\approx\) \(1.381015326\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.453505489276972732329133349760, −9.164357727767744996554627058200, −8.453234310603238542956108613739, −7.910181575788804740460685750914, −7.77698474257649382795973883033, −7.07715851976662932792536475690, −6.39550603345587366482416300943, −5.82638850103906378738309356342, −5.51933717252283675583847570281, −4.59219535120872901055676782946, −4.33214489367502481515592583866, −3.67417741905446743355217837016, −2.79160913843474065678897118940, −2.07080373981098315618079743058, −0.883484333734674189049376948209, 0.883484333734674189049376948209, 2.07080373981098315618079743058, 2.79160913843474065678897118940, 3.67417741905446743355217837016, 4.33214489367502481515592583866, 4.59219535120872901055676782946, 5.51933717252283675583847570281, 5.82638850103906378738309356342, 6.39550603345587366482416300943, 7.07715851976662932792536475690, 7.77698474257649382795973883033, 7.910181575788804740460685750914, 8.453234310603238542956108613739, 9.164357727767744996554627058200, 9.453505489276972732329133349760

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