Properties

Label 4-4031e2-1.1-c1e2-0-0
Degree $4$
Conductor $16248961$
Sign $1$
Analytic cond. $1036.04$
Root an. cond. $5.67342$
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  + 2·2-s + 2·3-s + 4-s + 4·5-s + 4·6-s − 2·7-s − 9-s + 8·10-s + 2·12-s + 10·13-s − 4·14-s + 8·15-s + 16-s + 6·17-s − 2·18-s − 2·19-s + 4·20-s − 4·21-s + 12·23-s + 2·25-s + 20·26-s − 6·27-s − 2·28-s + 2·29-s + 16·30-s + 4·31-s − 2·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.78·5-s + 1.63·6-s − 0.755·7-s − 1/3·9-s + 2.52·10-s + 0.577·12-s + 2.77·13-s − 1.06·14-s + 2.06·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s − 0.458·19-s + 0.894·20-s − 0.872·21-s + 2.50·23-s + 2/5·25-s + 3.92·26-s − 1.15·27-s − 0.377·28-s + 0.371·29-s + 2.92·30-s + 0.718·31-s − 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16248961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16248961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16248961\)    =    \(29^{2} \cdot 139^{2}\)
Sign: $1$
Analytic conductor: \(1036.04\)
Root analytic conductor: \(5.67342\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16248961,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(15.42757470\)
\(L(\frac12)\) \(\approx\) \(15.42757470\)
\(L(\frac{3}{2})\) 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)$
bad29$C_1$ \( ( 1 - T )^{2} \)
139$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 2 T + 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 105 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 28 T + 366 T^{2} + 28 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 201 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−8.511468521136550993252927669522, −8.445361455226514549143857845182, −8.044817370407442611537239136355, −7.51483316664061613205271064009, −6.95647873757739248726279849948, −6.64675502714306942259424180344, −6.06964454767894990182097768494, −6.04118977497786049401995843092, −5.68215834351972005705294834114, −5.29840142736284275786883936978, −5.05362436177255829529506537724, −4.31514482585859362862021316585, −3.80533769671276014998152164596, −3.78977069372423707882443427392, −3.01930448942650949189552672391, −2.92332821871196718639064209232, −2.68979780369886358348184156585, −1.66410408567194486387988019198, −1.50375815087984065269763681125, −0.851703007429084841432176246676, 0.851703007429084841432176246676, 1.50375815087984065269763681125, 1.66410408567194486387988019198, 2.68979780369886358348184156585, 2.92332821871196718639064209232, 3.01930448942650949189552672391, 3.78977069372423707882443427392, 3.80533769671276014998152164596, 4.31514482585859362862021316585, 5.05362436177255829529506537724, 5.29840142736284275786883936978, 5.68215834351972005705294834114, 6.04118977497786049401995843092, 6.06964454767894990182097768494, 6.64675502714306942259424180344, 6.95647873757739248726279849948, 7.51483316664061613205271064009, 8.044817370407442611537239136355, 8.445361455226514549143857845182, 8.511468521136550993252927669522

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