Properties

Label 4-21e4-1.1-c1e2-0-2
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
Motivic weight $1$
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  − 2-s + 2·4-s − 2·5-s − 5·8-s + 2·10-s + 4·11-s − 4·13-s + 5·16-s − 6·17-s − 4·19-s − 4·20-s − 4·22-s + 5·25-s + 4·26-s + 4·29-s − 10·32-s + 6·34-s − 6·37-s + 4·38-s + 10·40-s − 4·41-s − 8·43-s + 8·44-s − 5·50-s − 8·52-s + 6·53-s − 8·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 0.894·5-s − 1.76·8-s + 0.632·10-s + 1.20·11-s − 1.10·13-s + 5/4·16-s − 1.45·17-s − 0.917·19-s − 0.894·20-s − 0.852·22-s + 25-s + 0.784·26-s + 0.742·29-s − 1.76·32-s + 1.02·34-s − 0.986·37-s + 0.648·38-s + 1.58·40-s − 0.624·41-s − 1.21·43-s + 1.20·44-s − 0.707·50-s − 1.10·52-s + 0.824·53-s − 1.07·55-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6955965604\)
\(L(\frac12)\) \(\approx\) \(0.6955965604\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−11.80402897285165985849555844608, −10.81816057609690301350209065938, −10.61409736355128903825277258672, −9.863927074629101212979322956917, −9.613997248842167876723362570465, −8.785828459576528146697069600297, −8.736676061247533233536899379302, −8.448891794333314692240576878529, −7.63060349210109494651241503445, −7.05056013115321086583392089416, −6.85542351953130888969222340940, −6.38833289892065249201816713922, −5.98548577948485881999583093687, −4.82861947041117472738471645570, −4.79208794793551421877151840854, −3.56093595562485365691255699573, −3.55528996331510143687911431437, −2.38234167632467782041601104852, −2.06372222323479133124052619410, −0.56574615748523627800600037904, 0.56574615748523627800600037904, 2.06372222323479133124052619410, 2.38234167632467782041601104852, 3.55528996331510143687911431437, 3.56093595562485365691255699573, 4.79208794793551421877151840854, 4.82861947041117472738471645570, 5.98548577948485881999583093687, 6.38833289892065249201816713922, 6.85542351953130888969222340940, 7.05056013115321086583392089416, 7.63060349210109494651241503445, 8.448891794333314692240576878529, 8.736676061247533233536899379302, 8.785828459576528146697069600297, 9.613997248842167876723362570465, 9.863927074629101212979322956917, 10.61409736355128903825277258672, 10.81816057609690301350209065938, 11.80402897285165985849555844608

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