Properties

Label 4-41472-1.1-c1e2-0-1
Degree $4$
Conductor $41472$
Sign $1$
Analytic cond. $2.64429$
Root an. cond. $1.27519$
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  + 8·11-s − 4·13-s + 8·23-s − 2·25-s − 4·37-s − 8·47-s + 2·49-s − 4·61-s + 8·71-s + 12·73-s + 8·83-s + 12·97-s + 16·107-s + 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 2.41·11-s − 1.10·13-s + 1.66·23-s − 2/5·25-s − 0.657·37-s − 1.16·47-s + 2/7·49-s − 0.512·61-s + 0.949·71-s + 1.40·73-s + 0.878·83-s + 1.21·97-s + 1.54·107-s + 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41472\)    =    \(2^{9} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.64429\)
Root analytic conductor: \(1.27519\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 41472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.449646380\)
\(L(\frac12)\) \(\approx\) \(1.449646380\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−10.10415348120348902951298002782, −9.659624217350462529728144521941, −9.188781868703027942550187325661, −8.893403616743268785382876025504, −8.252648232278485430949911229152, −7.43904305346956878425952166602, −7.08110449917677950598863988254, −6.48562033737500993050832764104, −6.12316075571524015907000533390, −5.05570696136568803315394787787, −4.78053437666407406327353625842, −3.83030057795491944209525043594, −3.38801150536137460244799725913, −2.27036987463474694377407703426, −1.22265358585269878860427135405, 1.22265358585269878860427135405, 2.27036987463474694377407703426, 3.38801150536137460244799725913, 3.83030057795491944209525043594, 4.78053437666407406327353625842, 5.05570696136568803315394787787, 6.12316075571524015907000533390, 6.48562033737500993050832764104, 7.08110449917677950598863988254, 7.43904305346956878425952166602, 8.252648232278485430949911229152, 8.893403616743268785382876025504, 9.188781868703027942550187325661, 9.659624217350462529728144521941, 10.10415348120348902951298002782

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