Properties

Label 4-2028e2-1.1-c1e2-0-14
Degree $4$
Conductor $4112784$
Sign $1$
Analytic cond. $262.234$
Root an. cond. $4.02413$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 4·17-s − 6·25-s − 4·27-s − 12·29-s − 8·43-s + 10·49-s + 8·51-s − 20·53-s − 28·61-s + 12·75-s − 32·79-s + 5·81-s + 24·87-s − 20·101-s + 16·103-s + 24·107-s + 12·113-s + 6·121-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 20·147-s + 149-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 0.970·17-s − 6/5·25-s − 0.769·27-s − 2.22·29-s − 1.21·43-s + 10/7·49-s + 1.12·51-s − 2.74·53-s − 3.58·61-s + 1.38·75-s − 3.60·79-s + 5/9·81-s + 2.57·87-s − 1.99·101-s + 1.57·103-s + 2.32·107-s + 1.12·113-s + 6/11·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·147-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4112784\)    =    \(2^{4} \cdot 3^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(262.234\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4112784,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−8.868846233846508506388452521795, −8.813732844963216489184794391429, −7.935730992416060941358040400705, −7.80433978468021520564411993845, −7.21570051869570027856824075318, −7.12695148958217466605537996887, −6.47007881210334126460844651450, −5.98465857057036710103781826432, −5.92532997716241188678680318446, −5.48453599620452708294071401486, −4.79193386425839096467955534518, −4.58728859259556129333279487065, −4.17737100000295686817537665077, −3.54236846157479395782221387964, −3.17816743860810052265591214063, −2.35866999897939738587899993063, −1.72379456102557725178058192806, −1.42242276437796690933912837265, 0, 0, 1.42242276437796690933912837265, 1.72379456102557725178058192806, 2.35866999897939738587899993063, 3.17816743860810052265591214063, 3.54236846157479395782221387964, 4.17737100000295686817537665077, 4.58728859259556129333279487065, 4.79193386425839096467955534518, 5.48453599620452708294071401486, 5.92532997716241188678680318446, 5.98465857057036710103781826432, 6.47007881210334126460844651450, 7.12695148958217466605537996887, 7.21570051869570027856824075318, 7.80433978468021520564411993845, 7.935730992416060941358040400705, 8.813732844963216489184794391429, 8.868846233846508506388452521795

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