Properties

Degree $4$
Conductor $5308416$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·11-s + 4·13-s + 8·23-s + 8·25-s + 16·37-s − 24·47-s + 6·49-s + 16·61-s + 8·71-s − 16·73-s − 24·83-s + 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2.41·11-s + 1.10·13-s + 1.66·23-s + 8/5·25-s + 2.63·37-s − 3.50·47-s + 6/7·49-s + 2.04·61-s + 0.949·71-s − 1.87·73-s − 2.63·83-s + 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.155509198\)
\(L(\frac12)\) \(\approx\) \(4.155509198\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−9.182796358858131062005746422852, −8.671132159022916897196033894620, −8.539105003596820625964827664326, −8.352174433618556756146477667705, −7.49112417903938353981118073297, −7.26508398200434080060145066892, −6.65999770087998053024576615956, −6.62178011092317198411646866205, −6.19019991054668863312000503695, −5.81178760462294483788582839207, −5.18977828004671580578011363272, −4.74871547394549170034787114716, −4.31004380098127609361863092683, −4.01354161046819685313686401912, −3.34591889857637317195337084234, −3.16424180665631368706856515975, −2.52494194985834679533180471482, −1.66296134316450211293625620685, −1.16084756680104319818946279339, −0.876164686237853542763896459170, 0.876164686237853542763896459170, 1.16084756680104319818946279339, 1.66296134316450211293625620685, 2.52494194985834679533180471482, 3.16424180665631368706856515975, 3.34591889857637317195337084234, 4.01354161046819685313686401912, 4.31004380098127609361863092683, 4.74871547394549170034787114716, 5.18977828004671580578011363272, 5.81178760462294483788582839207, 6.19019991054668863312000503695, 6.62178011092317198411646866205, 6.65999770087998053024576615956, 7.26508398200434080060145066892, 7.49112417903938353981118073297, 8.352174433618556756146477667705, 8.539105003596820625964827664326, 8.671132159022916897196033894620, 9.182796358858131062005746422852

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