Properties

Label 4-112e3-1.1-c1e2-0-29
Degree $4$
Conductor $1404928$
Sign $-1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 2·11-s − 2·23-s + 6·25-s + 4·37-s − 2·43-s + 49-s − 2·53-s − 16·67-s − 16·71-s − 2·77-s − 12·79-s − 9·81-s + 12·107-s + 8·109-s + 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 2·161-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.603·11-s − 0.417·23-s + 6/5·25-s + 0.657·37-s − 0.304·43-s + 1/7·49-s − 0.274·53-s − 1.95·67-s − 1.89·71-s − 0.227·77-s − 1.35·79-s − 81-s + 1.16·107-s + 0.766·109-s + 0.752·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.157·161-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1404928\)    =    \(2^{12} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(89.5794\)
Root analytic conductor: \(3.07646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1404928,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 96 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
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

−7.72241116776539421853716100147, −7.39634895449226505323089333998, −6.87819101008056403094928120435, −6.46469819543302556769681526327, −5.84477785838798009633945034362, −5.65361191819037703152705874293, −5.00392542954768469100392585022, −4.54098447402841223153458187592, −4.29405581311628705713220084947, −3.51109404763741893260833273332, −2.94742977366229571348649529745, −2.58030816912804311934489283543, −1.76654589420157138744522952975, −1.14871450938102501502389666039, 0, 1.14871450938102501502389666039, 1.76654589420157138744522952975, 2.58030816912804311934489283543, 2.94742977366229571348649529745, 3.51109404763741893260833273332, 4.29405581311628705713220084947, 4.54098447402841223153458187592, 5.00392542954768469100392585022, 5.65361191819037703152705874293, 5.84477785838798009633945034362, 6.46469819543302556769681526327, 6.87819101008056403094928120435, 7.39634895449226505323089333998, 7.72241116776539421853716100147

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