Properties

Degree $4$
Conductor $194481$
Sign $1$
Motivic weight $1$
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: $1$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(2.59348\)
\(L(\frac12)\) \(\approx\) \(2.59348\)
\(L(\frac{3}{2})\) 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

−11.42246573592320040955126608197, −11.08816099052193730586026017416, −10.75716102976924003391464350569, −10.28656416713327785743945911650, −9.307113614828823043519987160241, −9.005376370676956116405339931168, −8.520270031082421588158231239740, −8.246147397636992144189467428597, −7.65630259062102721473457234019, −6.93298289875662016544307328450, −6.66328616045060353467405089350, −6.24781448814553709647427376173, −5.30207812062550686158929015907, −5.12967714732655062142470444546, −4.15787096548369957526349467029, −4.08922303615558275264143400460, −3.55027268374249635468919687494, −3.38373017834395905072088271255, −1.98209792667588231872305161133, −0.886873009659944755673861585793, 0.886873009659944755673861585793, 1.98209792667588231872305161133, 3.38373017834395905072088271255, 3.55027268374249635468919687494, 4.08922303615558275264143400460, 4.15787096548369957526349467029, 5.12967714732655062142470444546, 5.30207812062550686158929015907, 6.24781448814553709647427376173, 6.66328616045060353467405089350, 6.93298289875662016544307328450, 7.65630259062102721473457234019, 8.246147397636992144189467428597, 8.520270031082421588158231239740, 9.005376370676956116405339931168, 9.307113614828823043519987160241, 10.28656416713327785743945911650, 10.75716102976924003391464350569, 11.08816099052193730586026017416, 11.42246573592320040955126608197

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