Properties

Label 4-189e2-1.1-c1e2-0-2
Degree $4$
Conductor $35721$
Sign $1$
Analytic cond. $2.27760$
Root an. cond. $1.22848$
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-s − 5-s + 4·7-s + 8·8-s + 2·10-s + 5·11-s + 5·13-s − 8·14-s − 7·16-s + 3·17-s − 19-s + 20-s − 10·22-s + 3·23-s + 5·25-s − 10·26-s − 4·28-s − 29-s − 14·32-s − 6·34-s − 4·35-s − 3·37-s + 2·38-s − 8·40-s − 5·41-s + 43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 2.82·8-s + 0.632·10-s + 1.50·11-s + 1.38·13-s − 2.13·14-s − 7/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s − 2.13·22-s + 0.625·23-s + 25-s − 1.96·26-s − 0.755·28-s − 0.185·29-s − 2.47·32-s − 1.02·34-s − 0.676·35-s − 0.493·37-s + 0.324·38-s − 1.26·40-s − 0.780·41-s + 0.152·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(35721\)    =    \(3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.27760\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{189} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 35721,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6302383999\)
\(L(\frac12)\) \(\approx\) \(0.6302383999\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−12.56228453862573756095980609702, −12.46212104465039664491971004511, −11.41339285891468038713768079353, −11.32924784759997815795275612755, −10.61988458834588128445306079576, −10.51838487147393649811474419264, −9.527710891089699547149021574599, −9.254913218369034317099183957610, −8.787510548118890264028300112934, −8.355259586042305130754459436618, −8.041558117334159433696107494868, −7.51857919503984356996870283016, −6.85993090869678789422442766272, −6.09809757732958974580992158973, −5.01179670474333687540389094978, −4.84405173034031777728197531192, −3.98784000574745962223690651716, −3.55698322445557741853188652806, −1.54879141719047672260390690944, −1.15954005450070349770990608819, 1.15954005450070349770990608819, 1.54879141719047672260390690944, 3.55698322445557741853188652806, 3.98784000574745962223690651716, 4.84405173034031777728197531192, 5.01179670474333687540389094978, 6.09809757732958974580992158973, 6.85993090869678789422442766272, 7.51857919503984356996870283016, 8.041558117334159433696107494868, 8.355259586042305130754459436618, 8.787510548118890264028300112934, 9.254913218369034317099183957610, 9.527710891089699547149021574599, 10.51838487147393649811474419264, 10.61988458834588128445306079576, 11.32924784759997815795275612755, 11.41339285891468038713768079353, 12.46212104465039664491971004511, 12.56228453862573756095980609702

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