Properties

Label 4-442368-1.1-c1e2-0-62
Degree $4$
Conductor $442368$
Sign $-1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 8·19-s − 2·25-s + 27-s − 24·43-s + 6·49-s − 8·57-s + 8·67-s − 4·73-s − 2·75-s + 81-s − 28·97-s − 6·121-s + 127-s − 24·129-s + 131-s + 137-s + 139-s + 6·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s − 8·171-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.83·19-s − 2/5·25-s + 0.192·27-s − 3.65·43-s + 6/7·49-s − 1.05·57-s + 0.977·67-s − 0.468·73-s − 0.230·75-s + 1/9·81-s − 2.84·97-s − 0.545·121-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.494·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s − 0.611·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.276673959117272624117788538965, −8.172325717989392708885417424216, −7.49209581127155268652912226841, −6.86145520372086407937034011675, −6.66476390073654020746263143409, −6.14849922753886032508434138816, −5.48781805482364691047173427259, −4.99883221539357316029878248597, −4.42013999551616182274089412630, −3.92415158866083962323458195287, −3.42574309754453412717795186057, −2.71316566665094768593083864396, −2.09503952708704878470371581797, −1.47125558197166324196700994002, 0, 1.47125558197166324196700994002, 2.09503952708704878470371581797, 2.71316566665094768593083864396, 3.42574309754453412717795186057, 3.92415158866083962323458195287, 4.42013999551616182274089412630, 4.99883221539357316029878248597, 5.48781805482364691047173427259, 6.14849922753886032508434138816, 6.66476390073654020746263143409, 6.86145520372086407937034011675, 7.49209581127155268652912226841, 8.172325717989392708885417424216, 8.276673959117272624117788538965

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