Properties

Label 4-21e4-1.1-c1e2-0-16
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  + 2·2-s + 4-s + 4·5-s + 8·10-s + 4·11-s − 8·13-s + 16-s + 4·17-s + 4·20-s + 8·22-s + 4·23-s + 4·25-s − 16·26-s + 8·29-s + 8·31-s − 2·32-s + 8·34-s − 8·37-s + 4·41-s + 4·44-s + 8·46-s + 8·50-s − 8·52-s + 4·53-s + 16·55-s + 16·58-s − 8·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.78·5-s + 2.52·10-s + 1.20·11-s − 2.21·13-s + 1/4·16-s + 0.970·17-s + 0.894·20-s + 1.70·22-s + 0.834·23-s + 4/5·25-s − 3.13·26-s + 1.48·29-s + 1.43·31-s − 0.353·32-s + 1.37·34-s − 1.31·37-s + 0.624·41-s + 0.603·44-s + 1.17·46-s + 1.13·50-s − 1.10·52-s + 0.549·53-s + 2.15·55-s + 2.10·58-s − 1.04·59-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

\(L(1)\) \(\approx\) \(4.677308486\)
\(L(\frac12)\) \(\approx\) \(4.677308486\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 168 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 64 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 208 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−11.67783777386154003194467814591, −10.79481371538445815474399457382, −10.33521766744250026935737225966, −10.02993801833414807224646031324, −9.627229635967836355226680600683, −9.258967309191065318163505056602, −8.841193091559993307222322056928, −8.088777723242449878848007036447, −7.44680288581468260844254222535, −7.05205720589200451921747377812, −6.37768776651124213838204058241, −6.14117276544581238883924624336, −5.32873745649406525399312494406, −5.27009850611074602684499346558, −4.51681678749834850661437747057, −4.35394752223632909721139669670, −3.27754524893025953274612190053, −2.83488901717425392302673622849, −2.07454851217190769150187883915, −1.28901064488714939397850557810, 1.28901064488714939397850557810, 2.07454851217190769150187883915, 2.83488901717425392302673622849, 3.27754524893025953274612190053, 4.35394752223632909721139669670, 4.51681678749834850661437747057, 5.27009850611074602684499346558, 5.32873745649406525399312494406, 6.14117276544581238883924624336, 6.37768776651124213838204058241, 7.05205720589200451921747377812, 7.44680288581468260844254222535, 8.088777723242449878848007036447, 8.841193091559993307222322056928, 9.258967309191065318163505056602, 9.627229635967836355226680600683, 10.02993801833414807224646031324, 10.33521766744250026935737225966, 10.79481371538445815474399457382, 11.67783777386154003194467814591

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