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-s − 2·4-s + 4·5-s + 2·7-s − 3·8-s + 4·10-s − 8·13-s + 2·14-s + 16-s − 6·19-s − 8·20-s − 8·23-s + 7·25-s − 8·26-s − 4·28-s − 6·29-s + 2·32-s + 8·35-s − 2·37-s − 6·38-s − 12·40-s + 12·41-s − 4·43-s − 8·46-s + 6·47-s + 3·49-s + 7·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 1.78·5-s + 0.755·7-s − 1.06·8-s + 1.26·10-s − 2.21·13-s + 0.534·14-s + 1/4·16-s − 1.37·19-s − 1.78·20-s − 1.66·23-s + 7/5·25-s − 1.56·26-s − 0.755·28-s − 1.11·29-s + 0.353·32-s + 1.35·35-s − 0.328·37-s − 0.973·38-s − 1.89·40-s + 1.87·41-s − 0.609·43-s − 1.17·46-s + 0.875·47-s + 3/7·49-s + 0.989·50-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$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 241 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.63813437874785310287418557675, −7.26431258075961080975780352399, −7.18013865380870035451533553304, −6.40847190316644082491886012821, −6.08890221009689197627026391744, −5.89609301496905915405494362152, −5.54286685193393162016362741556, −5.32047121164360914964388329543, −4.77726091023720538350779707239, −4.53378299162898426102505629251, −4.24557051341291062627371141018, −4.11895612654706002560906920496, −3.17457543550619130841475540323, −3.02436133877598197689771950347, −2.22400806894733838308411387173, −2.07592259011891859731476305873, −1.89095327592551982788316470165, −1.11971564393352770182648888670, 0, 0, 1.11971564393352770182648888670, 1.89095327592551982788316470165, 2.07592259011891859731476305873, 2.22400806894733838308411387173, 3.02436133877598197689771950347, 3.17457543550619130841475540323, 4.11895612654706002560906920496, 4.24557051341291062627371141018, 4.53378299162898426102505629251, 4.77726091023720538350779707239, 5.32047121164360914964388329543, 5.54286685193393162016362741556, 5.89609301496905915405494362152, 6.08890221009689197627026391744, 6.40847190316644082491886012821, 7.18013865380870035451533553304, 7.26431258075961080975780352399, 7.63813437874785310287418557675

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