Properties

Label 4-231525-1.1-c1e2-0-36
Degree $4$
Conductor $231525$
Sign $-1$
Analytic cond. $14.7622$
Root an. cond. $1.96014$
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 + 2·5-s + 2·6-s − 7-s − 8·8-s + 9-s + 4·10-s − 12-s − 4·13-s − 2·14-s + 2·15-s − 7·16-s + 2·18-s − 2·20-s − 21-s − 8·24-s − 25-s − 8·26-s + 27-s + 28-s + 4·30-s + 14·32-s − 2·35-s − 36-s − 4·39-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 2.82·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 0.516·15-s − 7/4·16-s + 0.471·18-s − 0.447·20-s − 0.218·21-s − 1.63·24-s − 1/5·25-s − 1.56·26-s + 0.192·27-s + 0.188·28-s + 0.730·30-s + 2.47·32-s − 0.338·35-s − 1/6·36-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(231525\)    =    \(3^{3} \cdot 5^{2} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(14.7622\)
Root analytic conductor: \(1.96014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 231525,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−8.950452560934461228952758349280, −8.383266094837325727408472045987, −7.81295430367747747098376616892, −7.31923002930315464027308864727, −6.57215946307210072487478723379, −5.94933200323688252851139603477, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.81505610216268765778329020979, −4.32385123346365347175721472715, −3.64262157144388694664848575283, −3.05422074105458389777226041971, −2.69870051831794005458111315936, −1.67935906551934617681697486496, 0, 1.67935906551934617681697486496, 2.69870051831794005458111315936, 3.05422074105458389777226041971, 3.64262157144388694664848575283, 4.32385123346365347175721472715, 4.81505610216268765778329020979, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 5.94933200323688252851139603477, 6.57215946307210072487478723379, 7.31923002930315464027308864727, 7.81295430367747747098376616892, 8.383266094837325727408472045987, 8.950452560934461228952758349280

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