Properties

Label 4-7938e2-1.1-c1e2-0-9
Degree $4$
Conductor $63011844$
Sign $1$
Analytic cond. $4017.68$
Root an. cond. $7.96148$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 6·11-s + 2·13-s + 5·16-s + 6·17-s + 2·19-s + 12·22-s − 6·23-s − 7·25-s − 4·26-s − 6·29-s + 2·31-s − 6·32-s − 12·34-s + 4·37-s − 4·38-s + 12·41-s + 4·43-s − 18·44-s + 12·46-s − 6·47-s + 14·50-s + 6·52-s − 12·53-s + 12·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.80·11-s + 0.554·13-s + 5/4·16-s + 1.45·17-s + 0.458·19-s + 2.55·22-s − 1.25·23-s − 7/5·25-s − 0.784·26-s − 1.11·29-s + 0.359·31-s − 1.06·32-s − 2.05·34-s + 0.657·37-s − 0.648·38-s + 1.87·41-s + 0.609·43-s − 2.71·44-s + 1.76·46-s − 0.875·47-s + 1.97·50-s + 0.832·52-s − 1.64·53-s + 1.57·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(63011844\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4017.68\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 63011844,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 132 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 247 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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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

−7.74671721865724625637901407200, −7.63746402065382400153937252411, −7.32477436173228081224845685000, −6.68821685910079755402141530246, −6.22688449808974176248504567909, −5.95704212229321978482626246225, −5.65548500378443992368949053229, −5.59654489521095325153702553735, −4.75771356584939530616134999571, −4.68694700754188630326622835804, −3.90075934228242579549846329763, −3.59592596902147513212601137403, −3.19988105402462027607504491121, −2.77747775883350069513971786582, −2.16501366495056473369659127140, −2.15034526022237570402147650190, −1.28374999424732012170339337441, −1.05546918753098166008581357206, 0, 0, 1.05546918753098166008581357206, 1.28374999424732012170339337441, 2.15034526022237570402147650190, 2.16501366495056473369659127140, 2.77747775883350069513971786582, 3.19988105402462027607504491121, 3.59592596902147513212601137403, 3.90075934228242579549846329763, 4.68694700754188630326622835804, 4.75771356584939530616134999571, 5.59654489521095325153702553735, 5.65548500378443992368949053229, 5.95704212229321978482626246225, 6.22688449808974176248504567909, 6.68821685910079755402141530246, 7.32477436173228081224845685000, 7.63746402065382400153937252411, 7.74671721865724625637901407200

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