Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{2} \cdot 7^{4} $
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  + 2·3-s + 5-s + 3·9-s + 11-s − 5·13-s + 2·15-s − 8·17-s + 5·19-s + 8·23-s + 5·25-s + 4·27-s − 3·29-s − 2·31-s + 2·33-s − 3·37-s − 10·39-s + 6·41-s + 7·43-s + 3·45-s − 12·47-s − 16·51-s − 11·53-s + 55-s + 10·57-s − 5·59-s − 20·61-s − 5·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 9-s + 0.301·11-s − 1.38·13-s + 0.516·15-s − 1.94·17-s + 1.14·19-s + 1.66·23-s + 25-s + 0.769·27-s − 0.557·29-s − 0.359·31-s + 0.348·33-s − 0.493·37-s − 1.60·39-s + 0.937·41-s + 1.06·43-s + 0.447·45-s − 1.75·47-s − 2.24·51-s − 1.51·53-s + 0.134·55-s + 1.32·57-s − 0.650·59-s − 2.56·61-s − 0.620·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9408} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.784548797\)
\(L(\frac12)\)  \(\approx\)  \(3.784548797\)
\(L(\frac{3}{2})\)   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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + T + 132 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} \)
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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

−7.66328531982910988548536257136, −7.64992175954345097120988135535, −7.33126789711657316884512611365, −6.79917022602003021690101807611, −6.65380468460929706355078888468, −6.35837839237825744027627275949, −5.85410935915425778699537735219, −5.34953985618329060208957704906, −4.90378106355982411668098203641, −4.86044661479823303219741803988, −4.28217321843639640422893790064, −4.24471567835145342075536636457, −3.26464277823186490474947102453, −3.24488238917503924636695537243, −2.88047509841753621674424224774, −2.49571966549554907731089548591, −1.86465137484388749313564147516, −1.77154419026422537891331533282, −1.10682712970812488746740921956, −0.39781015255148828026636074956, 0.39781015255148828026636074956, 1.10682712970812488746740921956, 1.77154419026422537891331533282, 1.86465137484388749313564147516, 2.49571966549554907731089548591, 2.88047509841753621674424224774, 3.24488238917503924636695537243, 3.26464277823186490474947102453, 4.24471567835145342075536636457, 4.28217321843639640422893790064, 4.86044661479823303219741803988, 4.90378106355982411668098203641, 5.34953985618329060208957704906, 5.85410935915425778699537735219, 6.35837839237825744027627275949, 6.65380468460929706355078888468, 6.79917022602003021690101807611, 7.33126789711657316884512611365, 7.64992175954345097120988135535, 7.66328531982910988548536257136

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