Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{8} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 6·11-s − 2·13-s − 8·19-s − 6·23-s + 5·25-s + 6·29-s − 8·31-s + 4·37-s + 12·41-s + 4·43-s + 12·47-s + 12·53-s + 10·61-s − 8·67-s − 12·71-s − 20·73-s + 6·77-s + 4·79-s − 12·83-s − 24·89-s + 2·91-s + 10·97-s − 12·101-s − 8·103-s + 12·107-s + 28·109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.80·11-s − 0.554·13-s − 1.83·19-s − 1.25·23-s + 25-s + 1.11·29-s − 1.43·31-s + 0.657·37-s + 1.87·41-s + 0.609·43-s + 1.75·47-s + 1.64·53-s + 1.28·61-s − 0.977·67-s − 1.42·71-s − 2.34·73-s + 0.683·77-s + 0.450·79-s − 1.31·83-s − 2.54·89-s + 0.209·91-s + 1.01·97-s − 1.19·101-s − 0.788·103-s + 1.16·107-s + 2.68·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{2268} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6346130780\)
\(L(\frac12)\)  \(\approx\)  \(0.6346130780\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.954083983417968950778595705529, −8.714856379864109155626557695190, −8.691708748413608990135737158582, −8.025460003388224801738524162685, −7.58700247885894869043234030820, −7.34087334667278035791086019937, −7.06485535008402857043182799627, −6.45319460686884828453882081888, −5.94208133767132300212196196801, −5.69729846878226190519614019299, −5.47005332806805922470972451189, −4.66826700627741717419464487389, −4.35508326462730648850119497692, −4.15706293341624450597361385327, −3.43994453101223211628149269579, −2.61454819182830619596687862796, −2.59775249824722536351258407399, −2.19555793939704123652728816113, −1.20751825795538845446611096919, −0.28194619615959937496546034655, 0.28194619615959937496546034655, 1.20751825795538845446611096919, 2.19555793939704123652728816113, 2.59775249824722536351258407399, 2.61454819182830619596687862796, 3.43994453101223211628149269579, 4.15706293341624450597361385327, 4.35508326462730648850119497692, 4.66826700627741717419464487389, 5.47005332806805922470972451189, 5.69729846878226190519614019299, 5.94208133767132300212196196801, 6.45319460686884828453882081888, 7.06485535008402857043182799627, 7.34087334667278035791086019937, 7.58700247885894869043234030820, 8.025460003388224801738524162685, 8.691708748413608990135737158582, 8.714856379864109155626557695190, 8.954083983417968950778595705529

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