Properties

Degree $4$
Conductor $614656$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s − 14·19-s + 7·25-s + 14·27-s − 12·29-s − 10·31-s − 10·37-s − 6·47-s − 18·53-s + 28·57-s − 18·59-s − 14·75-s − 4·81-s − 24·83-s + 24·87-s + 20·93-s + 2·103-s + 22·109-s + 20·111-s + 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 12·141-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 3.21·19-s + 7/5·25-s + 2.69·27-s − 2.22·29-s − 1.79·31-s − 1.64·37-s − 0.875·47-s − 2.47·53-s + 3.70·57-s − 2.34·59-s − 1.61·75-s − 4/9·81-s − 2.63·83-s + 2.57·87-s + 2.07·93-s + 0.197·103-s + 2.10·109-s + 1.89·111-s + 1.12·113-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.01·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 107 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35604132320528957203170963425, −9.643056866037542507659769792663, −9.112175241764027222849099826751, −8.811387563063235717931825421809, −8.424139223477214800317221720020, −8.147538782730181230325227529983, −7.20485023568168009974022485961, −7.10152964215978916313235657320, −6.29576725602317367963590051517, −6.18300725804138709988492524008, −5.72246113567273475562701106067, −5.25680477505961232193244877553, −4.62396189926426788124148700380, −4.43140932129377275546216121694, −3.32709934635960876556728387188, −3.24897452515178296615306938291, −2.12425479085155313175740843581, −1.73270708803653011484508016380, 0, 0, 1.73270708803653011484508016380, 2.12425479085155313175740843581, 3.24897452515178296615306938291, 3.32709934635960876556728387188, 4.43140932129377275546216121694, 4.62396189926426788124148700380, 5.25680477505961232193244877553, 5.72246113567273475562701106067, 6.18300725804138709988492524008, 6.29576725602317367963590051517, 7.10152964215978916313235657320, 7.20485023568168009974022485961, 8.147538782730181230325227529983, 8.424139223477214800317221720020, 8.811387563063235717931825421809, 9.112175241764027222849099826751, 9.643056866037542507659769792663, 10.35604132320528957203170963425

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