Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 7^{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-s + 2·3-s − 5-s + 2·6-s − 4·7-s − 8-s + 3·9-s − 10-s − 3·11-s − 2·13-s − 4·14-s − 2·15-s − 16-s + 6·17-s + 3·18-s + 19-s − 8·21-s − 3·22-s − 9·23-s − 2·24-s − 2·26-s + 10·27-s + 12·29-s − 2·30-s − 8·31-s − 6·33-s + 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 0.447·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 9-s − 0.316·10-s − 0.904·11-s − 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.229·19-s − 1.74·21-s − 0.639·22-s − 1.87·23-s − 0.408·24-s − 0.392·26-s + 1.92·27-s + 2.22·29-s − 0.365·30-s − 1.43·31-s − 1.04·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{70} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 4900,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.24333\)
\(L(\frac12)\)  \(\approx\)  \(1.24333\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−14.61777141842794189768454698025, −14.57781183609616189848107520147, −13.71089637602596785375825156220, −13.68579527590435481843316327750, −12.59224917444173072857038973396, −12.50121732159152632150920036533, −12.24788227275347666382833646586, −11.12291794042038496292391479291, −10.19095008837039158422829506066, −9.948938616610033244479183380716, −9.433724438513604545475250100804, −8.617441261340900804509559804041, −7.84833271908281547941958143290, −7.60422602125420421213833691831, −6.55221263625625154346273123197, −5.95989562751509034618273960297, −4.92336720086814895226941027035, −4.06859797742588904077155461716, −3.20395034731392218686573802636, −2.73046000169629407206692505395, 2.73046000169629407206692505395, 3.20395034731392218686573802636, 4.06859797742588904077155461716, 4.92336720086814895226941027035, 5.95989562751509034618273960297, 6.55221263625625154346273123197, 7.60422602125420421213833691831, 7.84833271908281547941958143290, 8.617441261340900804509559804041, 9.433724438513604545475250100804, 9.948938616610033244479183380716, 10.19095008837039158422829506066, 11.12291794042038496292391479291, 12.24788227275347666382833646586, 12.50121732159152632150920036533, 12.59224917444173072857038973396, 13.68579527590435481843316327750, 13.71089637602596785375825156220, 14.57781183609616189848107520147, 14.61777141842794189768454698025

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