Properties

Label 4-847-1.1-c1e2-0-0
Degree $4$
Conductor $847$
Sign $1$
Analytic cond. $0.0540054$
Root an. cond. $0.482069$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·5-s − 7-s + 4·8-s − 5·9-s − 8·10-s + 2·14-s − 4·16-s − 8·17-s + 10·18-s + 2·19-s + 2·23-s + 3·25-s − 6·29-s + 12·31-s + 16·34-s − 4·35-s + 14·37-s − 4·38-s + 16·40-s − 2·41-s + 2·43-s − 20·45-s − 4·46-s + 8·47-s − 4·49-s − 6·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.78·5-s − 0.377·7-s + 1.41·8-s − 5/3·9-s − 2.52·10-s + 0.534·14-s − 16-s − 1.94·17-s + 2.35·18-s + 0.458·19-s + 0.417·23-s + 3/5·25-s − 1.11·29-s + 2.15·31-s + 2.74·34-s − 0.676·35-s + 2.30·37-s − 0.648·38-s + 2.52·40-s − 0.312·41-s + 0.304·43-s − 2.98·45-s − 0.589·46-s + 1.16·47-s − 4/7·49-s − 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.0540054\)
Root analytic conductor: \(0.482069\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 847,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2621188089\)
\(L(\frac12)\) \(\approx\) \(0.2621188089\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−19.4606214574, −19.1857249719, −18.2934649330, −17.9414335735, −17.5198844891, −17.0336103204, −16.9819106548, −15.9140726033, −15.1197748738, −14.2484377732, −13.6045057483, −13.5686390571, −12.7512335921, −11.4512586103, −11.0392861257, −10.0355090972, −9.59490413612, −9.12193960981, −8.60353961929, −7.79543673471, −6.36261389471, −5.89237731397, −4.69940224529, −2.52489627324, 2.52489627324, 4.69940224529, 5.89237731397, 6.36261389471, 7.79543673471, 8.60353961929, 9.12193960981, 9.59490413612, 10.0355090972, 11.0392861257, 11.4512586103, 12.7512335921, 13.5686390571, 13.6045057483, 14.2484377732, 15.1197748738, 15.9140726033, 16.9819106548, 17.0336103204, 17.5198844891, 17.9414335735, 18.2934649330, 19.1857249719, 19.4606214574

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