Properties

Label 4-21e4-1.1-c1e2-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·13-s − 3·16-s + 8·19-s + 2·25-s + 8·31-s + 4·37-s − 8·43-s + 4·52-s + 20·61-s + 7·64-s − 8·67-s − 28·73-s − 8·76-s + 16·79-s − 28·97-s − 2·100-s + 8·103-s + 4·109-s − 10·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.10·13-s − 3/4·16-s + 1.83·19-s + 2/5·25-s + 1.43·31-s + 0.657·37-s − 1.21·43-s + 0.554·52-s + 2.56·61-s + 7/8·64-s − 0.977·67-s − 3.27·73-s − 0.917·76-s + 1.80·79-s − 2.84·97-s − 1/5·100-s + 0.788·103-s + 0.383·109-s − 0.909·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.328·148-s + 0.0819·149-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

\(L(1)\) \(\approx\) \(1.291338071\)
\(L(\frac12)\) \(\approx\) \(1.291338071\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−11.48393794420244135371546013396, −10.95658054654902885523846772136, −10.17188386146879038957946866480, −10.06144849154688278201195905973, −9.508923167256671296794728048315, −9.268646300567467243411926807400, −8.588054341470238711310997797416, −8.225844865232840812413709801409, −7.69810934092326468321480626486, −7.11368080298575350370680898060, −6.86728429350633734444790732393, −6.21520433376882407027295525295, −5.34195569456028256586443128798, −5.29179036278425807274949680025, −4.45556336501987772086854948897, −4.23077073410533406981583562252, −3.13108407536332909137163043288, −2.85720623146223462659941831225, −1.88630160201182295709583291815, −0.74095505323005833418238937755, 0.74095505323005833418238937755, 1.88630160201182295709583291815, 2.85720623146223462659941831225, 3.13108407536332909137163043288, 4.23077073410533406981583562252, 4.45556336501987772086854948897, 5.29179036278425807274949680025, 5.34195569456028256586443128798, 6.21520433376882407027295525295, 6.86728429350633734444790732393, 7.11368080298575350370680898060, 7.69810934092326468321480626486, 8.225844865232840812413709801409, 8.588054341470238711310997797416, 9.268646300567467243411926807400, 9.508923167256671296794728048315, 10.06144849154688278201195905973, 10.17188386146879038957946866480, 10.95658054654902885523846772136, 11.48393794420244135371546013396

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