Properties

Degree $4$
Conductor $28901376$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 9-s − 12·17-s + 6·25-s − 4·41-s + 3·49-s − 2·63-s + 12·73-s − 32·79-s + 81-s + 28·89-s + 36·97-s − 16·103-s − 28·113-s − 24·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 0.755·7-s − 1/3·9-s − 2.91·17-s + 6/5·25-s − 0.624·41-s + 3/7·49-s − 0.251·63-s + 1.40·73-s − 3.60·79-s + 1/9·81-s + 2.96·89-s + 3.65·97-s − 1.57·103-s − 2.63·113-s − 2.20·119-s + 6/11·121-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 + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28901376\)    =    \(2^{16} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{5376} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28901376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628040871\)
\(L(\frac12)\) \(\approx\) \(1.628040871\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−8.350517693041277270735599658787, −8.229852418487187662755735810790, −7.44656064467612503377806834424, −7.40039972237262266761801591035, −6.91500810570979008521380896108, −6.53224710163648512416763596229, −6.29349622223517076572567579270, −5.98080186583108787327345899579, −5.21765163285796894274478632993, −5.12645213216649848307893433446, −4.72792771328861545304698602156, −4.32335758214917138159052045224, −4.05094925871767125867850116134, −3.52254486113530012337037724094, −2.88956951127148359134542731072, −2.64875407948134032120680640951, −1.93209494769392056603312768058, −1.91973623200180722729461837883, −1.04504733105666765977175647277, −0.36009807948245557570723673684, 0.36009807948245557570723673684, 1.04504733105666765977175647277, 1.91973623200180722729461837883, 1.93209494769392056603312768058, 2.64875407948134032120680640951, 2.88956951127148359134542731072, 3.52254486113530012337037724094, 4.05094925871767125867850116134, 4.32335758214917138159052045224, 4.72792771328861545304698602156, 5.12645213216649848307893433446, 5.21765163285796894274478632993, 5.98080186583108787327345899579, 6.29349622223517076572567579270, 6.53224710163648512416763596229, 6.91500810570979008521380896108, 7.40039972237262266761801591035, 7.44656064467612503377806834424, 8.229852418487187662755735810790, 8.350517693041277270735599658787

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