Properties

Label 4-21e4-1.1-c5e2-0-5
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $5002.62$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 9·4-s − 72·5-s + 15·8-s − 216·10-s − 480·11-s + 1.29e3·13-s − 529·16-s − 936·17-s − 216·19-s + 648·20-s − 1.44e3·22-s + 504·23-s − 2.36e3·25-s + 3.88e3·26-s − 6.37e3·29-s + 9.93e3·31-s − 5.79e3·32-s − 2.80e3·34-s + 1.11e4·37-s − 648·38-s − 1.08e3·40-s − 2.09e4·41-s − 6.26e3·43-s + 4.32e3·44-s + 1.51e3·46-s − 7.92e3·47-s + ⋯
L(s)  = 1  + 0.530·2-s − 0.281·4-s − 1.28·5-s + 0.0828·8-s − 0.683·10-s − 1.19·11-s + 2.12·13-s − 0.516·16-s − 0.785·17-s − 0.137·19-s + 0.362·20-s − 0.634·22-s + 0.198·23-s − 0.755·25-s + 1.12·26-s − 1.40·29-s + 1.85·31-s − 1.00·32-s − 0.416·34-s + 1.33·37-s − 0.0727·38-s − 0.106·40-s − 1.94·41-s − 0.516·43-s + 0.336·44-s + 0.105·46-s − 0.522·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5002.62\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 194481,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$D_{4}$ \( 1 - 3 T + 9 p T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \)
5$C_2$ \( ( 1 + 36 T + p^{5} T^{2} )^{2} \)
11$D_{4}$ \( 1 + 480 T + 376614 T^{2} + 480 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 1296 T + 912362 T^{2} - 1296 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 936 T + 807586 T^{2} + 936 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 216 T + 961814 T^{2} + 216 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 504 T + 12784878 T^{2} - 504 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 6372 T + 31409694 T^{2} + 6372 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 9936 T + 65931134 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 11124 T + 115818446 T^{2} - 11124 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 20952 T + 339207826 T^{2} + 20952 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 6264 T + 25906310 T^{2} + 6264 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 7920 T - 101923298 T^{2} + 7920 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 2220 T + 769983534 T^{2} + 2220 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 504 p T + 1614887590 T^{2} + 504 p^{6} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 17280 T + 707051402 T^{2} + 17280 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 20680 T + 11658642 p T^{2} + 20680 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 92280 T + 5423120334 T^{2} - 92280 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 56592 T + 4777720274 T^{2} - 56592 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 56096 T + 4914766302 T^{2} + 56096 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 71352 T + 4792627990 T^{2} - 71352 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 123192 T + 14311603186 T^{2} + 123192 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 35856 T - 1009376254 T^{2} - 35856 p^{5} T^{3} + p^{10} T^{4} \)
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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

−10.16796018554225294144428298000, −9.712709122393030863485025696978, −8.972605405864452239929932479973, −8.720666654300518761836006854296, −8.061160512203894297825318278694, −7.958810638465216044128767776331, −7.49154779494772117120694906896, −6.70410558402122803501774336586, −6.31474891901061091744784120983, −5.87979325166003528392137598957, −4.98152861759306868455910770502, −4.87383340436066500426176727243, −4.04871306593225445512755470686, −3.81958507626244868794461265985, −3.34968875311475324579506409503, −2.54203190746228453997094126595, −1.81855741798239935593978434442, −1.04691134463053307894317112920, 0, 0, 1.04691134463053307894317112920, 1.81855741798239935593978434442, 2.54203190746228453997094126595, 3.34968875311475324579506409503, 3.81958507626244868794461265985, 4.04871306593225445512755470686, 4.87383340436066500426176727243, 4.98152861759306868455910770502, 5.87979325166003528392137598957, 6.31474891901061091744784120983, 6.70410558402122803501774336586, 7.49154779494772117120694906896, 7.958810638465216044128767776331, 8.061160512203894297825318278694, 8.720666654300518761836006854296, 8.972605405864452239929932479973, 9.712709122393030863485025696978, 10.16796018554225294144428298000

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