Properties

Label 4-231868-1.1-c1e2-0-5
Degree $4$
Conductor $231868$
Sign $-1$
Analytic cond. $14.7841$
Root an. cond. $1.96086$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 7-s + 4·8-s − 5·9-s − 6·11-s + 2·14-s + 5·16-s − 10·18-s − 12·22-s − 6·23-s − 10·25-s + 3·28-s + 6·32-s − 15·36-s − 14·37-s + 16·43-s − 18·44-s − 12·46-s + 49-s − 20·50-s − 24·53-s + 4·56-s − 5·63-s + 7·64-s + 10·67-s + 24·71-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s − 5/3·9-s − 1.80·11-s + 0.534·14-s + 5/4·16-s − 2.35·18-s − 2.55·22-s − 1.25·23-s − 2·25-s + 0.566·28-s + 1.06·32-s − 5/2·36-s − 2.30·37-s + 2.43·43-s − 2.71·44-s − 1.76·46-s + 1/7·49-s − 2.82·50-s − 3.29·53-s + 0.534·56-s − 0.629·63-s + 7/8·64-s + 1.22·67-s + 2.84·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(231868\)    =    \(2^{2} \cdot 7^{3} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(14.7841\)
Root analytic conductor: \(1.96086\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 231868,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( 1 - T \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
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

−8.357542398475312637001956914796, −8.284684574830567460945123276136, −7.63380170955425435565378568582, −7.56044433828328293164495992231, −6.55543304367446106392236047763, −6.10743185313712551583625613621, −5.72895855893330977484350370148, −5.19256942155997860504028721022, −5.06509234546123458101387314677, −4.16724531605991793017887414807, −3.60858467418748472558474969025, −3.06437335560668526609552925191, −2.33000973337707432224237511883, −2.01051836387845799731692253165, 0, 2.01051836387845799731692253165, 2.33000973337707432224237511883, 3.06437335560668526609552925191, 3.60858467418748472558474969025, 4.16724531605991793017887414807, 5.06509234546123458101387314677, 5.19256942155997860504028721022, 5.72895855893330977484350370148, 6.10743185313712551583625613621, 6.55543304367446106392236047763, 7.56044433828328293164495992231, 7.63380170955425435565378568582, 8.284684574830567460945123276136, 8.357542398475312637001956914796

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