Properties

Degree 4
Conductor $ 61^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 6·7-s − 3·8-s + 6·9-s + 9·10-s − 16·12-s + 2·13-s + 18·14-s + 12·15-s + 3·16-s − 12·17-s − 18·18-s + 2·19-s − 12·20-s + 24·21-s + 12·24-s + 5·25-s − 6·26-s + 4·27-s − 24·28-s − 3·29-s − 36·30-s − 18·31-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 2.26·7-s − 1.06·8-s + 2·9-s + 2.84·10-s − 4.61·12-s + 0.554·13-s + 4.81·14-s + 3.09·15-s + 3/4·16-s − 2.91·17-s − 4.24·18-s + 0.458·19-s − 2.68·20-s + 5.23·21-s + 2.44·24-s + 25-s − 1.17·26-s + 0.769·27-s − 4.53·28-s − 0.557·29-s − 6.57·30-s − 3.23·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3721\)    =    \(61^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3721} (1, \cdot )$
Sato-Tate  :  $E_6$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 3721,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 61$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 61$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad61$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−18.4028704621, −17.8311365782, −17.6116772369, −16.7569635712, −16.6641176394, −16.0654258792, −15.9584244754, −15.4539863608, −14.4970335637, −13.0066910864, −13.0036758549, −12.3577327492, −11.4902501698, −11.2217542815, −10.8637892872, −10.3086549347, −9.486771795, −8.93412491479, −8.68715484894, −7.32351152279, −7.08316854214, −6.17446311784, −5.92144739335, −4.55343076694, −3.39144853322, 0, 0, 3.39144853322, 4.55343076694, 5.92144739335, 6.17446311784, 7.08316854214, 7.32351152279, 8.68715484894, 8.93412491479, 9.486771795, 10.3086549347, 10.8637892872, 11.2217542815, 11.4902501698, 12.3577327492, 13.0036758549, 13.0066910864, 14.4970335637, 15.4539863608, 15.9584244754, 16.0654258792, 16.6641176394, 16.7569635712, 17.6116772369, 17.8311365782, 18.4028704621

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