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 + 2·4-s + 4·5-s − 2·7-s − 4·8-s − 8·10-s + 2·13-s + 4·14-s + 8·16-s + 4·17-s + 6·19-s + 8·20-s − 10·23-s + 5·25-s − 4·26-s − 4·28-s − 6·29-s + 8·31-s − 8·32-s − 8·34-s − 8·35-s − 12·37-s − 12·38-s − 16·40-s − 8·41-s − 6·43-s + 20·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s − 0.755·7-s − 1.41·8-s − 2.52·10-s + 0.554·13-s + 1.06·14-s + 2·16-s + 0.970·17-s + 1.37·19-s + 1.78·20-s − 2.08·23-s + 25-s − 0.784·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s − 1.41·32-s − 1.37·34-s − 1.35·35-s − 1.97·37-s − 1.94·38-s − 2.52·40-s − 1.24·41-s − 0.914·43-s + 2.94·46-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$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 83 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 6 T + 4 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 168 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 199 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 216 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−7.63904652206657779081458307581, −7.56281493288620956313322006548, −7.00975015173377592437661304972, −6.64698133836767640949023354429, −6.28061646112836672212387731178, −5.92805958479976010042178001096, −5.83302423305163685350634578220, −5.53339743087358398309661294709, −5.10373080308573760173316594488, −4.63266477778450701817866466001, −3.86387575841633374180560709588, −3.41902814943474054848937196136, −3.39543698488040164789938736829, −2.72284744247203233236987706637, −2.40510663050330155554048137452, −1.80965353932400059676799728791, −1.36079094486045536322818369327, −1.21064718299637104702911621695, 0, 0, 1.21064718299637104702911621695, 1.36079094486045536322818369327, 1.80965353932400059676799728791, 2.40510663050330155554048137452, 2.72284744247203233236987706637, 3.39543698488040164789938736829, 3.41902814943474054848937196136, 3.86387575841633374180560709588, 4.63266477778450701817866466001, 5.10373080308573760173316594488, 5.53339743087358398309661294709, 5.83302423305163685350634578220, 5.92805958479976010042178001096, 6.28061646112836672212387731178, 6.64698133836767640949023354429, 7.00975015173377592437661304972, 7.56281493288620956313322006548, 7.63904652206657779081458307581

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