Properties

Label 4-126e2-1.1-c1e2-0-6
Degree $4$
Conductor $15876$
Sign $1$
Analytic cond. $1.01226$
Root an. cond. $1.00305$
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-s + 3·5-s − 7-s − 8-s − 3·9-s + 3·10-s + 6·11-s − 2·13-s − 14-s − 16-s + 12·17-s − 3·18-s − 14·19-s + 6·22-s − 3·23-s + 5·25-s − 2·26-s − 6·29-s − 2·31-s + 12·34-s − 3·35-s + 4·37-s − 14·38-s − 3·40-s − 2·43-s − 9·45-s − 3·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.948·10-s + 1.80·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 2.91·17-s − 0.707·18-s − 3.21·19-s + 1.27·22-s − 0.625·23-s + 25-s − 0.392·26-s − 1.11·29-s − 0.359·31-s + 2.05·34-s − 0.507·35-s + 0.657·37-s − 2.27·38-s − 0.474·40-s − 0.304·43-s − 1.34·45-s − 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15876\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.01226\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15876,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690079146\)
\(L(\frac12)\) \(\approx\) \(1.690079146\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−13.65357224935547719695164869503, −13.20986910823196139391488959493, −12.54574775076249188563250726737, −12.24928004782496096231071561576, −11.88455747256029342077452353981, −11.06723454414901334002775045985, −10.52514854254777462354520007841, −9.865036300491147798837126901256, −9.578776388011096610624079001792, −8.943665352280116993837185077328, −8.475795705608970126586706196731, −7.72947248319515276352626506245, −6.71548736137882161307211737010, −6.34169381378257625831735145687, −5.61153587545137261138159368982, −5.58530102299595835378930166780, −4.28245679157121827719439111253, −3.75638913511956709659578632254, −2.78406084968574424703432640976, −1.77013939322723170388853964880, 1.77013939322723170388853964880, 2.78406084968574424703432640976, 3.75638913511956709659578632254, 4.28245679157121827719439111253, 5.58530102299595835378930166780, 5.61153587545137261138159368982, 6.34169381378257625831735145687, 6.71548736137882161307211737010, 7.72947248319515276352626506245, 8.475795705608970126586706196731, 8.943665352280116993837185077328, 9.578776388011096610624079001792, 9.865036300491147798837126901256, 10.52514854254777462354520007841, 11.06723454414901334002775045985, 11.88455747256029342077452353981, 12.24928004782496096231071561576, 12.54574775076249188563250726737, 13.20986910823196139391488959493, 13.65357224935547719695164869503

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