# Properties

 Degree 4 Conductor $11^{2} \cdot 29^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 + 2·5-s − 4·7-s − 5·9-s + 8·13-s − 4·16-s − 2·23-s − 7·25-s − 8·35-s − 10·45-s − 2·49-s − 12·53-s + 10·59-s + 20·63-s + 16·65-s − 14·67-s − 6·71-s − 8·80-s + 16·81-s − 12·83-s − 32·91-s − 32·103-s + 36·107-s + 20·109-s + 16·112-s − 4·115-s − 40·117-s + 121-s + ⋯
 L(s)  = 1 + 0.894·5-s − 1.51·7-s − 5/3·9-s + 2.21·13-s − 16-s − 0.417·23-s − 7/5·25-s − 1.35·35-s − 1.49·45-s − 2/7·49-s − 1.64·53-s + 1.30·59-s + 2.51·63-s + 1.98·65-s − 1.71·67-s − 0.712·71-s − 0.894·80-s + 16/9·81-s − 1.31·83-s − 3.35·91-s − 3.15·103-s + 3.48·107-s + 1.91·109-s + 1.51·112-s − 0.373·115-s − 3.69·117-s + 1/11·121-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$101761$$    =    $$11^{2} \cdot 29^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{101761} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = 1 Selberg data = $(4,\ 101761,\ (\ :1/2, 1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{11,\;29\}$, $F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{11,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad11$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
29$C_2$ $$1 + p T^{2}$$
good2$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
5$C_2$ $$( 1 - T + p T^{2} )^{2}$$
7$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$ $$( 1 + p T^{2} )^{2}$$
23$C_2$ $$( 1 + T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
37$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
41$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
43$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
47$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
53$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
61$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
67$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
79$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
83$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
89$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
97$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
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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

−9.314615238921567438303356334000, −8.751241727709668354321300909980, −8.603539619290756001226038948684, −7.972705235455589069488978218046, −7.23499358099781255352016467414, −6.36261389471308870138602900888, −6.26725427945844283760086667796, −5.92310508625439103207168037966, −5.45933868177711943400258230362, −4.48608615943364982740005877095, −3.66126078491527460999719563871, −3.26980135440747142850588612235, −2.57349767368819643760419670867, −1.67480393474155854801846646588, 0, 1.67480393474155854801846646588, 2.57349767368819643760419670867, 3.26980135440747142850588612235, 3.66126078491527460999719563871, 4.48608615943364982740005877095, 5.45933868177711943400258230362, 5.92310508625439103207168037966, 6.26725427945844283760086667796, 6.36261389471308870138602900888, 7.23499358099781255352016467414, 7.972705235455589069488978218046, 8.603539619290756001226038948684, 8.751241727709668354321300909980, 9.314615238921567438303356334000