Properties

Label 4-1331712-1.1-c1e2-0-12
Degree $4$
Conductor $1331712$
Sign $1$
Analytic cond. $84.9111$
Root an. cond. $3.03557$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·9-s + 8·11-s + 2·17-s + 8·19-s − 6·25-s + 4·27-s + 16·33-s − 12·41-s + 8·43-s + 2·49-s + 4·51-s + 16·57-s − 8·59-s − 24·67-s + 20·73-s − 12·75-s + 5·81-s + 8·83-s − 12·89-s − 12·97-s + 24·99-s + 40·107-s − 12·113-s + 26·121-s − 24·123-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 2.41·11-s + 0.485·17-s + 1.83·19-s − 6/5·25-s + 0.769·27-s + 2.78·33-s − 1.87·41-s + 1.21·43-s + 2/7·49-s + 0.560·51-s + 2.11·57-s − 1.04·59-s − 2.93·67-s + 2.34·73-s − 1.38·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s − 1.21·97-s + 2.41·99-s + 3.86·107-s − 1.12·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1331712\)    =    \(2^{9} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(84.9111\)
Root analytic conductor: \(3.03557\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1331712,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.578657960\)
\(L(\frac12)\) \(\approx\) \(4.578657960\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−7.916184610763501150764377161438, −7.47440021870706367601671353104, −7.31889878147217285432019463149, −6.73870618720653177566812234969, −6.26895088903419553976746164792, −5.91991403599991886044725193544, −5.27442398825750717482876776028, −4.75760569558457030429860030016, −4.01215376994088253352189829152, −3.94512410946611821372106342064, −3.18935016627371752862222737803, −3.09370925732333153615131862731, −2.02316387502247347434526246665, −1.55655451926421595753779072230, −0.980696887521417523960583270574, 0.980696887521417523960583270574, 1.55655451926421595753779072230, 2.02316387502247347434526246665, 3.09370925732333153615131862731, 3.18935016627371752862222737803, 3.94512410946611821372106342064, 4.01215376994088253352189829152, 4.75760569558457030429860030016, 5.27442398825750717482876776028, 5.91991403599991886044725193544, 6.26895088903419553976746164792, 6.73870618720653177566812234969, 7.31889878147217285432019463149, 7.47440021870706367601671353104, 7.916184610763501150764377161438

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