Properties

Degree 4
Conductor $ 29^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4-s − 2·5-s − 4·6-s − 9-s + 4·10-s + 2·11-s + 2·12-s − 2·13-s − 4·15-s + 16-s − 4·17-s + 2·18-s + 12·19-s − 2·20-s − 4·22-s − 4·23-s − 7·25-s + 4·26-s − 6·27-s + 2·29-s + 8·30-s + 6·31-s + 2·32-s + 4·33-s + 8·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 1/3·9-s + 1.26·10-s + 0.603·11-s + 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 2.75·19-s − 0.447·20-s − 0.852·22-s − 0.834·23-s − 7/5·25-s + 0.784·26-s − 1.15·27-s + 0.371·29-s + 1.46·30-s + 1.07·31-s + 0.353·32-s + 0.696·33-s + 1.37·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(841\)    =    \(29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{841} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 841,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2915215656$
$L(\frac12)$  $\approx$  $0.2915215656$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 29$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 29$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad29$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$V_4$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 77 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$V_4$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$C_4$ \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.9066660622, −19.5460681837, −19.4812084888, −18.739827221, −17.9503494765, −17.5864978016, −17.3134398624, −16.0588838804, −15.9380677394, −15.2322435831, −14.4520003304, −13.8165445548, −13.6673297667, −12.0823286665, −11.937823971, −11.1688997427, −10.0425250247, −9.34700888024, −9.18473349505, −8.11169477787, −8.01335053311, −7.11407190599, −5.7119310376, −4.12707082881, −2.92477068592, 2.92477068592, 4.12707082881, 5.7119310376, 7.11407190599, 8.01335053311, 8.11169477787, 9.18473349505, 9.34700888024, 10.0425250247, 11.1688997427, 11.937823971, 12.0823286665, 13.6673297667, 13.8165445548, 14.4520003304, 15.2322435831, 15.9380677394, 16.0588838804, 17.3134398624, 17.5864978016, 17.9503494765, 18.739827221, 19.4812084888, 19.5460681837, 19.9066660622

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