Properties

Label 4-1632e2-1.1-c1e2-0-11
Degree $4$
Conductor $2663424$
Sign $1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
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  − 2·3-s + 9-s + 6·11-s + 6·17-s + 4·19-s − 2·25-s + 4·27-s − 12·33-s − 4·43-s − 2·49-s − 12·51-s − 8·57-s − 12·59-s + 16·67-s + 12·73-s + 4·75-s − 11·81-s + 12·83-s − 12·89-s + 12·97-s + 6·99-s + 18·107-s − 12·113-s + 14·121-s + 127-s + 8·129-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.80·11-s + 1.45·17-s + 0.917·19-s − 2/5·25-s + 0.769·27-s − 2.08·33-s − 0.609·43-s − 2/7·49-s − 1.68·51-s − 1.05·57-s − 1.56·59-s + 1.95·67-s + 1.40·73-s + 0.461·75-s − 1.22·81-s + 1.31·83-s − 1.27·89-s + 1.21·97-s + 0.603·99-s + 1.74·107-s − 1.12·113-s + 1.27·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.779085085\)
\(L(\frac12)\) \(\approx\) \(1.779085085\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−7.52747090377821956610605153883, −7.12594626527130568256562505024, −6.61273368560251435639710672123, −6.34133790600429928122816209747, −6.01179034260327607721608996457, −5.49023190868270211652990996329, −5.13846928602031464304778746189, −4.77032540662069658036349049952, −4.14461274373158340937124725800, −3.55649290189302970556073129086, −3.41687728026496463079874815945, −2.63330803552518618426178196434, −1.74655107964577286764543823969, −1.21695768216628226144600501045, −0.64615290850334764621603315778, 0.64615290850334764621603315778, 1.21695768216628226144600501045, 1.74655107964577286764543823969, 2.63330803552518618426178196434, 3.41687728026496463079874815945, 3.55649290189302970556073129086, 4.14461274373158340937124725800, 4.77032540662069658036349049952, 5.13846928602031464304778746189, 5.49023190868270211652990996329, 6.01179034260327607721608996457, 6.34133790600429928122816209747, 6.61273368560251435639710672123, 7.12594626527130568256562505024, 7.52747090377821956610605153883

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