Properties

Label 4-16875-1.1-c1e2-0-0
Degree $4$
Conductor $16875$
Sign $1$
Analytic cond. $1.07596$
Root an. cond. $1.01847$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s + 9-s − 3·12-s + 4·13-s + 5·16-s + 8·19-s + 27-s − 3·36-s + 20·37-s + 4·39-s − 8·43-s + 5·48-s − 14·49-s − 12·52-s + 8·57-s − 4·61-s − 3·64-s − 24·67-s − 20·73-s − 24·76-s + 81-s − 4·97-s + 32·103-s − 3·108-s + 28·109-s + 20·111-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s + 1/3·9-s − 0.866·12-s + 1.10·13-s + 5/4·16-s + 1.83·19-s + 0.192·27-s − 1/2·36-s + 3.28·37-s + 0.640·39-s − 1.21·43-s + 0.721·48-s − 2·49-s − 1.66·52-s + 1.05·57-s − 0.512·61-s − 3/8·64-s − 2.93·67-s − 2.34·73-s − 2.75·76-s + 1/9·81-s − 0.406·97-s + 3.15·103-s − 0.288·108-s + 2.68·109-s + 1.89·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16875\)    =    \(3^{3} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.07596\)
Root analytic conductor: \(1.01847\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{16875} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16875,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.032626388\)
\(L(\frac12)\) \(\approx\) \(1.032626388\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−11.19897971912574422599178738873, −10.04517050900761317536685250745, −9.965958896202437987484594622667, −9.327820441539330420804608673051, −8.843956595114680827452383410006, −8.455181921164028207665051683361, −7.68003844385765985083943135134, −7.46261028773657489832215471574, −6.20681526904542088835730053039, −5.86497653842965752491328535183, −4.79492945983632365647604814265, −4.52160444884874669934496061646, −3.55714635244582915245852508954, −3.01549784140686353642765162463, −1.27025970316177507740293768624, 1.27025970316177507740293768624, 3.01549784140686353642765162463, 3.55714635244582915245852508954, 4.52160444884874669934496061646, 4.79492945983632365647604814265, 5.86497653842965752491328535183, 6.20681526904542088835730053039, 7.46261028773657489832215471574, 7.68003844385765985083943135134, 8.455181921164028207665051683361, 8.843956595114680827452383410006, 9.327820441539330420804608673051, 9.965958896202437987484594622667, 10.04517050900761317536685250745, 11.19897971912574422599178738873

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