Properties

Label 4-243675-1.1-c1e2-0-2
Degree $4$
Conductor $243675$
Sign $-1$
Analytic cond. $15.5369$
Root an. cond. $1.98536$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s − 4-s + 2·6-s + 8·8-s + 9-s + 12-s − 7·16-s − 2·18-s + 4·19-s − 8·24-s + 25-s − 27-s − 4·29-s − 14·32-s − 36-s − 8·38-s + 20·41-s + 8·43-s + 7·48-s − 14·49-s − 2·50-s − 20·53-s + 2·54-s − 4·57-s + 8·58-s − 8·59-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.816·6-s + 2.82·8-s + 1/3·9-s + 0.288·12-s − 7/4·16-s − 0.471·18-s + 0.917·19-s − 1.63·24-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 2.47·32-s − 1/6·36-s − 1.29·38-s + 3.12·41-s + 1.21·43-s + 1.01·48-s − 2·49-s − 0.282·50-s − 2.74·53-s + 0.272·54-s − 0.529·57-s + 1.05·58-s − 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(243675\)    =    \(3^{3} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(15.5369\)
Root analytic conductor: \(1.98536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 243675,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.896447809747983381121393959041, −8.232011963291614168935511350860, −7.77071280677534797445852255244, −7.66488013441745380243230842523, −7.11524013136289051004798684067, −6.34145167556094658057083324143, −5.83679390833615760339265246745, −5.23920392624592057055772361749, −4.72109247762374245491183934060, −4.33195666036117235086666220277, −3.72492686467354363460541892516, −2.85380672919944569836723726469, −1.66341921070628603647910394136, −1.05029361783004892799606717843, 0, 1.05029361783004892799606717843, 1.66341921070628603647910394136, 2.85380672919944569836723726469, 3.72492686467354363460541892516, 4.33195666036117235086666220277, 4.72109247762374245491183934060, 5.23920392624592057055772361749, 5.83679390833615760339265246745, 6.34145167556094658057083324143, 7.11524013136289051004798684067, 7.66488013441745380243230842523, 7.77071280677534797445852255244, 8.232011963291614168935511350860, 8.896447809747983381121393959041

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