Properties

Label 4-2592e2-1.1-c1e2-0-30
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 6·13-s − 4·17-s + 5·25-s − 10·29-s − 4·37-s + 10·41-s + 7·49-s − 28·53-s + 10·61-s + 12·65-s − 12·73-s + 8·85-s − 20·89-s − 18·97-s − 2·101-s + 12·109-s − 14·113-s + 11·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.66·13-s − 0.970·17-s + 25-s − 1.85·29-s − 0.657·37-s + 1.56·41-s + 49-s − 3.84·53-s + 1.28·61-s + 1.48·65-s − 1.40·73-s + 0.867·85-s − 2.11·89-s − 1.82·97-s − 0.199·101-s + 1.14·109-s − 1.31·113-s + 121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
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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

−8.643592875549318202947190257217, −8.322147529835774295258516687193, −7.78945872239334069432400870310, −7.59770541603171538995517470939, −7.13284103448834137323053168164, −7.02116844692040028354355110798, −6.44597021406819334855555102172, −6.00013225684640725720855123249, −5.50467918575583979232735301346, −5.12197879180080150589992337435, −4.61522750701452278382929100407, −4.42553277927238074097862508984, −3.85023463839598664574010788232, −3.51015662420482243419935972866, −2.70080902117197413099128563541, −2.62637205874520343167341067342, −1.87434943159650414500715140578, −1.24908843676189811929719496116, 0, 0, 1.24908843676189811929719496116, 1.87434943159650414500715140578, 2.62637205874520343167341067342, 2.70080902117197413099128563541, 3.51015662420482243419935972866, 3.85023463839598664574010788232, 4.42553277927238074097862508984, 4.61522750701452278382929100407, 5.12197879180080150589992337435, 5.50467918575583979232735301346, 6.00013225684640725720855123249, 6.44597021406819334855555102172, 7.02116844692040028354355110798, 7.13284103448834137323053168164, 7.59770541603171538995517470939, 7.78945872239334069432400870310, 8.322147529835774295258516687193, 8.643592875549318202947190257217

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