Properties

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

Origins

Origins of factors

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Normalization:  

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

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