Properties

Degree $4$
Conductor $47753$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 2·7-s + 2·11-s − 3·16-s − 3·17-s + 6·25-s + 2·28-s + 10·29-s + 2·37-s + 4·43-s + 2·44-s − 10·49-s + 4·53-s + 6·59-s − 7·64-s − 3·68-s + 4·77-s − 9·81-s − 12·89-s − 2·97-s + 6·100-s + 10·107-s − 6·112-s − 10·113-s + 10·116-s − 6·119-s + 6·121-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s + 0.603·11-s − 3/4·16-s − 0.727·17-s + 6/5·25-s + 0.377·28-s + 1.85·29-s + 0.328·37-s + 0.609·43-s + 0.301·44-s − 1.42·49-s + 0.549·53-s + 0.781·59-s − 7/8·64-s − 0.363·68-s + 0.455·77-s − 81-s − 1.27·89-s − 0.203·97-s + 3/5·100-s + 0.966·107-s − 0.566·112-s − 0.940·113-s + 0.928·116-s − 0.550·119-s + 6/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47753 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47753 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47753\)    =    \(17 \cdot 53^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{47753} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 47753,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.680346598\)
\(L(\frac12)\) \(\approx\) \(1.680346598\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 132 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20031176998892011704503486271, −9.628381701329349363614943833849, −9.035598713483186025412233495073, −8.482936895576042873208258119215, −8.291486593957400338561786433882, −7.37702429122021546111228115401, −6.96661898125283035040685418532, −6.48096804345269936261233278524, −5.98795864547802251275846896815, −5.00869791480462797115881239936, −4.65686567470925407206120683820, −4.01792877493683480786978440635, −2.97069046567786444971729335240, −2.32576461206720290241880861530, −1.27908973548718496722710023918, 1.27908973548718496722710023918, 2.32576461206720290241880861530, 2.97069046567786444971729335240, 4.01792877493683480786978440635, 4.65686567470925407206120683820, 5.00869791480462797115881239936, 5.98795864547802251275846896815, 6.48096804345269936261233278524, 6.96661898125283035040685418532, 7.37702429122021546111228115401, 8.291486593957400338561786433882, 8.482936895576042873208258119215, 9.035598713483186025412233495073, 9.628381701329349363614943833849, 10.20031176998892011704503486271

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