Properties

Label 4-41472-1.1-c1e2-0-0
Degree $4$
Conductor $41472$
Sign $1$
Analytic cond. $2.64429$
Root an. cond. $1.27519$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·17-s + 16·23-s − 6·25-s + 16·31-s + 12·41-s − 14·49-s − 16·71-s + 20·73-s − 16·79-s + 12·89-s + 4·97-s + 32·103-s − 36·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.970·17-s + 3.33·23-s − 6/5·25-s + 2.87·31-s + 1.87·41-s − 2·49-s − 1.89·71-s + 2.34·73-s − 1.80·79-s + 1.27·89-s + 0.406·97-s + 3.15·103-s − 3.38·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 41472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.339615804\)
\(L(\frac12)\) \(\approx\) \(1.339615804\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.16546244400809085298891139382, −9.780091138821383013974264992430, −9.065170023749915638598404079220, −8.884556993307409273860919130264, −8.157389656224392853374347080198, −7.62403207119791885504595168937, −7.06857822018944724974086155331, −6.38250152569089720234352942221, −6.14969218484336320739052245813, −4.99175853408267534988469991920, −4.83326855247570334999188527992, −4.02227786472289347756607333391, −3.04433904291477058087427796742, −2.51659494204614894722398853037, −1.12835632773744922111480113749, 1.12835632773744922111480113749, 2.51659494204614894722398853037, 3.04433904291477058087427796742, 4.02227786472289347756607333391, 4.83326855247570334999188527992, 4.99175853408267534988469991920, 6.14969218484336320739052245813, 6.38250152569089720234352942221, 7.06857822018944724974086155331, 7.62403207119791885504595168937, 8.157389656224392853374347080198, 8.884556993307409273860919130264, 9.065170023749915638598404079220, 9.780091138821383013974264992430, 10.16546244400809085298891139382

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