Properties

Label 4-21e4-1.1-c2e2-0-2
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $144.393$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·3-s + 2·4-s − 6·5-s − 9·6-s + 3·8-s + 18·10-s − 3·11-s + 6·12-s − 4·13-s − 18·15-s − 3·16-s − 22·19-s − 12·20-s + 9·22-s − 48·23-s + 9·24-s − 25-s + 12·26-s − 27·27-s + 78·29-s + 54·30-s + 32·31-s + 12·32-s − 9·33-s − 68·37-s + 66·38-s + ⋯
L(s)  = 1  − 3/2·2-s + 3-s + 1/2·4-s − 6/5·5-s − 3/2·6-s + 3/8·8-s + 9/5·10-s − 0.272·11-s + 1/2·12-s − 0.307·13-s − 6/5·15-s − 0.187·16-s − 1.15·19-s − 3/5·20-s + 9/22·22-s − 2.08·23-s + 3/8·24-s − 0.0399·25-s + 6/13·26-s − 27-s + 2.68·29-s + 9/5·30-s + 1.03·31-s + 3/8·32-s − 0.272·33-s − 1.83·37-s + 1.73·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(144.393\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1654706685\)
\(L(\frac12)\) \(\approx\) \(0.1654706685\)
\(L(2)\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 3 T + 124 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2^2$ \( 1 + 4 T - 153 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 335 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 48 T + 1297 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 32 T + 63 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 21 T + 1828 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
47$C_2^2$ \( 1 - 84 T + 4561 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_2^2$ \( 1 + 87 T + 6004 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 56 T - 585 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9110 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 65 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 115 T + 3816 T^{2} + 115 p^{2} T^{3} + p^{4} T^{4} \)
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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

−10.85365429903368899682285319279, −10.37088454649139998465147996706, −10.22715415544686596910180834666, −9.745409316788149028732132121849, −9.060951390931452355521991858415, −8.847846943763536574564492713897, −8.357742853069927303723483973323, −8.191906145016284461504504462791, −7.66887425646739885631631514050, −7.53208387473437966742227349165, −6.57457084138698420347722117444, −6.23376129906621284834969570181, −5.46204472324425008586287315441, −4.61153630673139266162736789340, −4.06655112045638403606975738569, −3.83239087844297726504003168005, −2.65926086902810485770814681841, −2.54086999980883624873988362677, −1.31628506109092044566817564008, −0.22025839297479836410119128447, 0.22025839297479836410119128447, 1.31628506109092044566817564008, 2.54086999980883624873988362677, 2.65926086902810485770814681841, 3.83239087844297726504003168005, 4.06655112045638403606975738569, 4.61153630673139266162736789340, 5.46204472324425008586287315441, 6.23376129906621284834969570181, 6.57457084138698420347722117444, 7.53208387473437966742227349165, 7.66887425646739885631631514050, 8.191906145016284461504504462791, 8.357742853069927303723483973323, 8.847846943763536574564492713897, 9.060951390931452355521991858415, 9.745409316788149028732132121849, 10.22715415544686596910180834666, 10.37088454649139998465147996706, 10.85365429903368899682285319279

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