Properties

Degree $4$
Conductor $2822400$
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  − 3-s + 5-s − 4·7-s − 11-s + 2·13-s − 15-s − 3·19-s + 4·21-s + 7·23-s + 27-s − 16·29-s − 2·31-s + 33-s − 4·35-s − 11·37-s − 2·39-s − 22·41-s − 16·43-s − 5·47-s + 9·49-s + 11·53-s − 55-s + 3·57-s + 4·59-s + 2·65-s − 7·69-s + 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.688·19-s + 0.872·21-s + 1.45·23-s + 0.192·27-s − 2.97·29-s − 0.359·31-s + 0.174·33-s − 0.676·35-s − 1.80·37-s − 0.320·39-s − 3.43·41-s − 2.43·43-s − 0.729·47-s + 9/7·49-s + 1.51·53-s − 0.134·55-s + 0.397·57-s + 0.520·59-s + 0.248·65-s − 0.842·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2822400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2822400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04841415895\)
\(L(\frac12)\) \(\approx\) \(0.04841415895\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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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

−9.645529194206775674237179521887, −9.309689410272044474232605311289, −8.666328106996505670956017161449, −8.363259098663911736200420303808, −8.312083123535375178383890198474, −7.08918743127239478044254391619, −7.07423613665469414586408719283, −6.89001979491523585099234649437, −6.41807598244263102116197279392, −5.70158173067212268174121192934, −5.67792637778453247867409500125, −5.04168140523126230106896089692, −4.92545889399787184421972231256, −3.78222559471431756922802333872, −3.64969016491551752799892052416, −3.30921537606468101755556365988, −2.63533721770063924908287739903, −1.84697273236623692140284270232, −1.49137808468798994103135524713, −0.084000489316477839434140689195, 0.084000489316477839434140689195, 1.49137808468798994103135524713, 1.84697273236623692140284270232, 2.63533721770063924908287739903, 3.30921537606468101755556365988, 3.64969016491551752799892052416, 3.78222559471431756922802333872, 4.92545889399787184421972231256, 5.04168140523126230106896089692, 5.67792637778453247867409500125, 5.70158173067212268174121192934, 6.41807598244263102116197279392, 6.89001979491523585099234649437, 7.07423613665469414586408719283, 7.08918743127239478044254391619, 8.312083123535375178383890198474, 8.363259098663911736200420303808, 8.666328106996505670956017161449, 9.309689410272044474232605311289, 9.645529194206775674237179521887

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