Properties

Degree 4
Conductor $ 2^{2} \cdot 3 \cdot 41^{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  − 2·2-s − 3·3-s + 3·4-s − 4·5-s + 6·6-s − 2·7-s − 4·8-s + 4·9-s + 8·10-s − 6·11-s − 9·12-s + 4·14-s + 12·15-s + 5·16-s − 4·17-s − 8·18-s − 2·19-s − 12·20-s + 6·21-s + 12·22-s − 4·23-s + 12·24-s + 2·25-s − 6·28-s − 8·29-s − 24·30-s − 4·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 3/2·4-s − 1.78·5-s + 2.44·6-s − 0.755·7-s − 1.41·8-s + 4/3·9-s + 2.52·10-s − 1.80·11-s − 2.59·12-s + 1.06·14-s + 3.09·15-s + 5/4·16-s − 0.970·17-s − 1.88·18-s − 0.458·19-s − 2.68·20-s + 1.30·21-s + 2.55·22-s − 0.834·23-s + 2.44·24-s + 2/5·25-s − 1.13·28-s − 1.48·29-s − 4.38·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20172 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20172 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(20172\)    =    \(2^{2} \cdot 3 \cdot 41^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{20172} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 20172,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;41\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
41$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−16.1440817264, −16.0980463378, −15.5438761265, −15.2322872584, −14.8034113429, −13.5603503236, −13.1338784795, −12.3632258252, −12.3535797825, −11.5374061458, −11.2937627155, −10.8687735754, −10.5945207017, −9.81085245169, −9.48966114694, −8.46774633173, −8.02435745103, −7.75573819994, −6.85698182335, −6.7074892518, −5.69856539644, −5.36274716754, −4.29114537424, −3.54075497824, −2.32002779008, 0, 0, 2.32002779008, 3.54075497824, 4.29114537424, 5.36274716754, 5.69856539644, 6.7074892518, 6.85698182335, 7.75573819994, 8.02435745103, 8.46774633173, 9.48966114694, 9.81085245169, 10.5945207017, 10.8687735754, 11.2937627155, 11.5374061458, 12.3535797825, 12.3632258252, 13.1338784795, 13.5603503236, 14.8034113429, 15.2322872584, 15.5438761265, 16.0980463378, 16.1440817264

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