Properties

Degree 4
Conductor $ 3^{4} \cdot 7^{2} \cdot 11^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4-s − 5-s − 2·7-s + 8·8-s + 2·10-s − 2·13-s + 4·14-s − 7·16-s + 9·17-s − 3·19-s + 20-s − 5·23-s + 2·25-s + 4·26-s + 2·28-s − 4·29-s + 31-s − 14·32-s − 18·34-s + 2·35-s − 13·37-s + 6·38-s − 8·40-s − 15·41-s − 2·43-s + 10·46-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 2.82·8-s + 0.632·10-s − 0.554·13-s + 1.06·14-s − 7/4·16-s + 2.18·17-s − 0.688·19-s + 0.223·20-s − 1.04·23-s + 2/5·25-s + 0.784·26-s + 0.377·28-s − 0.742·29-s + 0.179·31-s − 2.47·32-s − 3.08·34-s + 0.338·35-s − 2.13·37-s + 0.973·38-s − 1.26·40-s − 2.34·41-s − 0.304·43-s + 1.47·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(58110129\)    =    \(3^{4} \cdot 7^{2} \cdot 11^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 58110129,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 13 T + 85 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 15 T + 127 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 150 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 167 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−7.66853476741148730248572221043, −7.66340059004799438838515002534, −7.18044414593632609202247797407, −6.88603948178923700006979658522, −6.35557063876303611309578568204, −6.14632144519866587344601277051, −5.34707741832851384457919340182, −5.22653768660608240930313322800, −5.02293874384626147573678822029, −4.55880766849690184720141595047, −3.94203535381861160856702839641, −3.74501388039750696524107084598, −3.32231877676726239253614514322, −3.17463509154787482144334220198, −2.15231024873850373622560961908, −1.83374285155447229927655523625, −1.36174737978857535738788788923, −0.77285977765209532936188379400, 0, 0, 0.77285977765209532936188379400, 1.36174737978857535738788788923, 1.83374285155447229927655523625, 2.15231024873850373622560961908, 3.17463509154787482144334220198, 3.32231877676726239253614514322, 3.74501388039750696524107084598, 3.94203535381861160856702839641, 4.55880766849690184720141595047, 5.02293874384626147573678822029, 5.22653768660608240930313322800, 5.34707741832851384457919340182, 6.14632144519866587344601277051, 6.35557063876303611309578568204, 6.88603948178923700006979658522, 7.18044414593632609202247797407, 7.66340059004799438838515002534, 7.66853476741148730248572221043

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