Properties

Label 4-672e2-1.1-c1e2-0-37
Degree $4$
Conductor $451584$
Sign $1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
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  − 3-s + 7-s − 2·11-s + 10·13-s + 2·17-s − 3·19-s − 21-s − 2·23-s + 5·25-s + 27-s + 16·29-s + 31-s + 2·33-s + 5·37-s − 10·39-s + 4·41-s + 14·43-s + 8·47-s − 6·49-s − 2·51-s + 2·53-s + 3·57-s − 10·59-s + 2·61-s + 11·67-s + 2·69-s + 24·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 0.603·11-s + 2.77·13-s + 0.485·17-s − 0.688·19-s − 0.218·21-s − 0.417·23-s + 25-s + 0.192·27-s + 2.97·29-s + 0.179·31-s + 0.348·33-s + 0.821·37-s − 1.60·39-s + 0.624·41-s + 2.13·43-s + 1.16·47-s − 6/7·49-s − 0.280·51-s + 0.274·53-s + 0.397·57-s − 1.30·59-s + 0.256·61-s + 1.34·67-s + 0.240·69-s + 2.84·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{672} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.004324095\)
\(L(\frac12)\) \(\approx\) \(2.004324095\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−10.83664555508901704742747250637, −10.58213763554481124016131105448, −9.790521770964003399701452324252, −9.633269741072575811828574383452, −8.744874791082726011338919275599, −8.495147862261865503692863698604, −8.228142734744287722552979378006, −7.916296783857286601235545036002, −6.89022464677755318365992258003, −6.83672102077519144216207351768, −6.05069579034749864449501904981, −5.92968103104968451139313600533, −5.43432835245337445491537850177, −4.72466592405287022646759610789, −4.15373004040385001547008461257, −3.93544302651761366684786896053, −2.83137774342708755548891080847, −2.68689873440682736571877829636, −1.30865545830965751972136080827, −0.956869995153323757616530987507, 0.956869995153323757616530987507, 1.30865545830965751972136080827, 2.68689873440682736571877829636, 2.83137774342708755548891080847, 3.93544302651761366684786896053, 4.15373004040385001547008461257, 4.72466592405287022646759610789, 5.43432835245337445491537850177, 5.92968103104968451139313600533, 6.05069579034749864449501904981, 6.83672102077519144216207351768, 6.89022464677755318365992258003, 7.916296783857286601235545036002, 8.228142734744287722552979378006, 8.495147862261865503692863698604, 8.744874791082726011338919275599, 9.633269741072575811828574383452, 9.790521770964003399701452324252, 10.58213763554481124016131105448, 10.83664555508901704742747250637

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