Properties

Label 4-21e4-1.1-c1e2-0-8
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  + 4-s + 3·5-s − 3·9-s + 3·11-s − 3·13-s − 3·16-s + 3·17-s + 9·19-s + 3·20-s + 9·23-s + 5·25-s − 9·29-s − 3·36-s − 7·37-s + 3·41-s − 43-s + 3·44-s − 9·45-s − 3·52-s + 15·53-s + 9·55-s − 7·64-s − 9·65-s − 8·67-s + 3·68-s + 9·73-s + 9·76-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.34·5-s − 9-s + 0.904·11-s − 0.832·13-s − 3/4·16-s + 0.727·17-s + 2.06·19-s + 0.670·20-s + 1.87·23-s + 25-s − 1.67·29-s − 1/2·36-s − 1.15·37-s + 0.468·41-s − 0.152·43-s + 0.452·44-s − 1.34·45-s − 0.416·52-s + 2.06·53-s + 1.21·55-s − 7/8·64-s − 1.11·65-s − 0.977·67-s + 0.363·68-s + 1.05·73-s + 1.03·76-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

\(L(1)\) \(\approx\) \(2.489375031\)
\(L(\frac12)\) \(\approx\) \(2.489375031\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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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

−11.40632432733225108182000205329, −10.92516608470411466306902315438, −10.50328884458875860361275540767, −9.861328490816099585712786417665, −9.367751134025422031806863187691, −9.347695112760174966469296447698, −8.841072049975956968599589901107, −8.240041641280609738669055701456, −7.37493504212361249897340938760, −7.20026455941060238563394181494, −6.80947594156380770543209143256, −6.06458625881309539539596987073, −5.63165539068504427278782897037, −5.25768247432087086051004324024, −4.87761163420525156942783335090, −3.78191481176377482405687732830, −3.17040035285208912741599036083, −2.64444307993536269557638908476, −1.94081127087227006539882941294, −1.08390794351730416373759210551, 1.08390794351730416373759210551, 1.94081127087227006539882941294, 2.64444307993536269557638908476, 3.17040035285208912741599036083, 3.78191481176377482405687732830, 4.87761163420525156942783335090, 5.25768247432087086051004324024, 5.63165539068504427278782897037, 6.06458625881309539539596987073, 6.80947594156380770543209143256, 7.20026455941060238563394181494, 7.37493504212361249897340938760, 8.240041641280609738669055701456, 8.841072049975956968599589901107, 9.347695112760174966469296447698, 9.367751134025422031806863187691, 9.861328490816099585712786417665, 10.50328884458875860361275540767, 10.92516608470411466306902315438, 11.40632432733225108182000205329

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