Properties

Label 4-21e4-1.1-c3e2-0-9
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  + 2·2-s + 8·4-s − 7·5-s + 40·8-s − 14·10-s − 5·11-s + 28·13-s + 80·16-s + 21·17-s + 49·19-s − 56·20-s − 10·22-s − 159·23-s + 125·25-s + 56·26-s − 116·29-s + 147·31-s + 320·32-s + 42·34-s − 219·37-s + 98·38-s − 280·40-s + 700·41-s − 248·43-s − 40·44-s − 318·46-s − 525·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s − 0.626·5-s + 1.76·8-s − 0.442·10-s − 0.137·11-s + 0.597·13-s + 5/4·16-s + 0.299·17-s + 0.591·19-s − 0.626·20-s − 0.0969·22-s − 1.44·23-s + 25-s + 0.422·26-s − 0.742·29-s + 0.851·31-s + 1.76·32-s + 0.211·34-s − 0.973·37-s + 0.418·38-s − 1.10·40-s + 2.66·41-s − 0.879·43-s − 0.137·44-s − 1.01·46-s − 1.62·47-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

\(L(2)\) \(\approx\) \(5.292907335\)
\(L(\frac12)\) \(\approx\) \(5.292907335\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - p T - p^{2} T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 5 T - 1306 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 14 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 21 T - 4472 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 49 T - 4458 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 159 T + 13114 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 147 T - 8182 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 219 T - 2692 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 350 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 124 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 525 T + 171802 T^{2} + 525 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 303 T - 57068 T^{2} - 303 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 105 T - 194354 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 413 T - 56412 T^{2} + 413 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 415 T - 128538 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 432 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1113 T + 849752 T^{2} + 1113 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 103 T - 482430 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 1092 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 329 T - 596728 T^{2} - 329 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 882 T + p^{3} T^{2} )^{2} \)
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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

−11.03926514914109270674773030661, −10.52422566606867945048235042321, −10.25196773183186935003118302094, −9.757853118851677833741115583815, −9.095710999589291826325230425474, −8.536604207943450782541083358859, −7.898108879276438483713970952290, −7.74575439156514383218661395330, −7.28735193023372115126915209954, −6.71947185453467222733909125214, −6.16104310389652089570864482625, −5.83583982496086992867010804308, −4.89261457818669842278298096500, −4.80494699966361850064616208640, −3.94303951354760821856082956948, −3.63497980941682298939895597841, −2.91462460474436470892873893938, −2.16309725505555033018869092385, −1.49809922045833177963939657630, −0.68262892341191589181056911257, 0.68262892341191589181056911257, 1.49809922045833177963939657630, 2.16309725505555033018869092385, 2.91462460474436470892873893938, 3.63497980941682298939895597841, 3.94303951354760821856082956948, 4.80494699966361850064616208640, 4.89261457818669842278298096500, 5.83583982496086992867010804308, 6.16104310389652089570864482625, 6.71947185453467222733909125214, 7.28735193023372115126915209954, 7.74575439156514383218661395330, 7.898108879276438483713970952290, 8.536604207943450782541083358859, 9.095710999589291826325230425474, 9.757853118851677833741115583815, 10.25196773183186935003118302094, 10.52422566606867945048235042321, 11.03926514914109270674773030661

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