Properties

Label 4-21e4-1.1-c3e2-0-7
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $677.032$
Root an. cond. $5.10096$
Motivic weight $3$
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-s + 8·4-s − 12·5-s − 23·8-s + 12·10-s + 20·11-s + 168·13-s + 23·16-s + 96·17-s + 12·19-s − 96·20-s − 20·22-s − 176·23-s + 125·25-s − 168·26-s − 116·29-s − 264·31-s − 184·32-s − 96·34-s − 258·37-s − 12·38-s + 276·40-s + 312·43-s + 160·44-s + 176·46-s + 408·47-s − 125·50-s + ⋯
L(s)  = 1  − 0.353·2-s + 4-s − 1.07·5-s − 1.01·8-s + 0.379·10-s + 0.548·11-s + 3.58·13-s + 0.359·16-s + 1.36·17-s + 0.144·19-s − 1.07·20-s − 0.193·22-s − 1.59·23-s + 25-s − 1.26·26-s − 0.742·29-s − 1.52·31-s − 1.01·32-s − 0.484·34-s − 1.14·37-s − 0.0512·38-s + 1.09·40-s + 1.10·43-s + 0.548·44-s + 0.564·46-s + 1.26·47-s − 0.353·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(677.032\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.232731086\)
\(L(\frac12)\) \(\approx\) \(2.232731086\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 12 T + 19 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 20 T - 931 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 84 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 96 T + 4303 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 12 T - 6715 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 176 T + 18809 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 264 T + 39905 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 258 T + 15911 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 156 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 408 T + 62641 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 722 T + 372407 T^{2} + 722 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 492 T + 36685 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 492 T + 15083 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 412 T - 131019 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 296 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 240 T - 331417 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 776 T + 109137 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 924 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 744 T - 151433 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 168 T + p^{3} T^{2} )^{2} \)
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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

−11.26944873593136688638968412223, −10.51580462255697629261295100852, −10.39507997126225145771758139324, −9.175128984338288616165063575190, −9.168505813606015775572158865010, −8.759274460311659251883494434803, −8.133220958571307050313269714085, −7.60246869281686540254680720112, −7.56152412251260532234993786401, −6.57476235179646911101079991341, −6.33001992566556794931794026253, −5.82736808873831508691561200318, −5.57144650156254054005810475612, −4.36641492013872712747760207101, −3.76933066749669812179729223869, −3.48641735823991290355648251467, −3.11590869055043645064590088894, −1.79590207195838378719183194059, −1.44176641217406390349784094451, −0.53682872322282660209933771217, 0.53682872322282660209933771217, 1.44176641217406390349784094451, 1.79590207195838378719183194059, 3.11590869055043645064590088894, 3.48641735823991290355648251467, 3.76933066749669812179729223869, 4.36641492013872712747760207101, 5.57144650156254054005810475612, 5.82736808873831508691561200318, 6.33001992566556794931794026253, 6.57476235179646911101079991341, 7.56152412251260532234993786401, 7.60246869281686540254680720112, 8.133220958571307050313269714085, 8.759274460311659251883494434803, 9.168505813606015775572158865010, 9.175128984338288616165063575190, 10.39507997126225145771758139324, 10.51580462255697629261295100852, 11.26944873593136688638968412223

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