Properties

Label 4-3822e2-1.1-c1e2-0-7
Degree $4$
Conductor $14607684$
Sign $1$
Analytic cond. $931.398$
Root an. cond. $5.52438$
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 + 3·4-s + 5-s + 4·6-s + 4·8-s + 3·9-s + 2·10-s + 5·11-s + 6·12-s − 2·13-s + 2·15-s + 5·16-s − 17-s + 6·18-s − 5·19-s + 3·20-s + 10·22-s + 3·23-s + 8·24-s + 25-s − 4·26-s + 4·27-s − 29-s + 4·30-s + 6·32-s + 10·33-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s + 1.41·8-s + 9-s + 0.632·10-s + 1.50·11-s + 1.73·12-s − 0.554·13-s + 0.516·15-s + 5/4·16-s − 0.242·17-s + 1.41·18-s − 1.14·19-s + 0.670·20-s + 2.13·22-s + 0.625·23-s + 1.63·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s − 0.185·29-s + 0.730·30-s + 1.06·32-s + 1.74·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14607684\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(931.398\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14607684,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(16.58979019\)
\(L(\frac12)\) \(\approx\) \(16.58979019\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 132 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 9 T + 156 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.792089130817399362758465445364, −8.251352455244099970065662024134, −7.73548167499260295173700254230, −7.59584046041812661538650921092, −7.14448322871862587834218805482, −6.75608033364335078939690047310, −6.34529538197202045315364619397, −6.16744775757243342477793843212, −5.74292246227952023492002227871, −5.25722507828402728268668980639, −4.62120120212975969738603863326, −4.42456190823828619421873725895, −4.12674046533058564668665622287, −3.77339413742182160705513686241, −3.20125577427704920430159034803, −2.80037503134609174492594909662, −2.31748354193214101954152805758, −2.12843806786283010744424863656, −1.38638969543504239388096097368, −0.890237578005539139561295283597, 0.890237578005539139561295283597, 1.38638969543504239388096097368, 2.12843806786283010744424863656, 2.31748354193214101954152805758, 2.80037503134609174492594909662, 3.20125577427704920430159034803, 3.77339413742182160705513686241, 4.12674046533058564668665622287, 4.42456190823828619421873725895, 4.62120120212975969738603863326, 5.25722507828402728268668980639, 5.74292246227952023492002227871, 6.16744775757243342477793843212, 6.34529538197202045315364619397, 6.75608033364335078939690047310, 7.14448322871862587834218805482, 7.59584046041812661538650921092, 7.73548167499260295173700254230, 8.251352455244099970065662024134, 8.792089130817399362758465445364

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