Properties

Degree 4
Conductor $ 11^{2} \cdot 89^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 5·9-s − 8·10-s + 2·11-s − 4·16-s − 4·17-s + 20·18-s + 16·20-s − 8·22-s − 7·25-s + 32·32-s + 16·34-s − 40·36-s − 16·40-s + 16·44-s − 10·45-s + 16·47-s − 10·49-s + 28·50-s − 12·53-s + 4·55-s − 64·64-s − 14·67-s − 32·68-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 5/3·9-s − 2.52·10-s + 0.603·11-s − 16-s − 0.970·17-s + 4.71·18-s + 3.57·20-s − 1.70·22-s − 7/5·25-s + 5.65·32-s + 2.74·34-s − 6.66·36-s − 2.52·40-s + 2.41·44-s − 1.49·45-s + 2.33·47-s − 1.42·49-s + 3.95·50-s − 1.64·53-s + 0.539·55-s − 8·64-s − 1.71·67-s − 3.88·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 958441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 958441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(958441\)    =    \(11^{2} \cdot 89^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{958441} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 958441,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.1707542378\)
\(L(\frac12)\)  \(\approx\)  \(0.1707542378\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{11,\;89\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{11,\;89\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad11$C_1$ \( ( 1 - T )^{2} \)
89$C_2$ \( 1 - 15 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.558580597072866362021424590520, −7.80306712748985717398688888740, −7.52305447162640040957017397420, −7.20685073138161009710557405792, −6.36261389471308870138602900888, −6.22751027744569053248642726916, −5.85551354442718043458262245689, −5.05232126574041880500064099657, −4.54859976719023207502696657866, −3.88151059584158277500212230098, −2.98547319486052976896224337133, −2.35691778866774912809470218958, −1.93214884663950861864873375229, −1.33739722426120305081140732346, −0.31591320242433503945261590043, 0.31591320242433503945261590043, 1.33739722426120305081140732346, 1.93214884663950861864873375229, 2.35691778866774912809470218958, 2.98547319486052976896224337133, 3.88151059584158277500212230098, 4.54859976719023207502696657866, 5.05232126574041880500064099657, 5.85551354442718043458262245689, 6.22751027744569053248642726916, 6.36261389471308870138602900888, 7.20685073138161009710557405792, 7.52305447162640040957017397420, 7.80306712748985717398688888740, 8.558580597072866362021424590520

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