Properties

Degree 4
Conductor $ 43^{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 − 3-s − 4-s + 5-s − 2·6-s − 3·7-s − 8·8-s + 3·9-s + 2·10-s + 12-s + 5·13-s − 6·14-s − 15-s − 7·16-s − 3·17-s + 6·18-s − 19-s − 20-s + 3·21-s + 7·23-s + 8·24-s + 5·25-s + 10·26-s − 8·27-s + 3·28-s − 3·29-s − 2·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s − 1.13·7-s − 2.82·8-s + 9-s + 0.632·10-s + 0.288·12-s + 1.38·13-s − 1.60·14-s − 0.258·15-s − 7/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s + 1.45·23-s + 1.63·24-s + 25-s + 1.96·26-s − 1.53·27-s + 0.566·28-s − 0.557·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1849\)    =    \(43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1849,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.781927\)
\(L(\frac12)\)  \(\approx\)  \(0.781927\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad43$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−16.25893957648634954297268744775, −15.48698718155727381100159608334, −14.93259322616907854553657847490, −14.68264660679604815517282447134, −13.52877102884076407184858132043, −13.25989587995379888651411950046, −12.98060071194356131373890504545, −12.83759326146348050290394206512, −11.62116892256875750743888082634, −11.32610889315071847199510580771, −10.00079602267096372848508341429, −9.864497437365040705722664129784, −8.848449434936333430085111719598, −8.577406316803923368617198473940, −6.76486336296373504490087508573, −6.52279276308513502587000590701, −5.51855702707420985771365031225, −5.03108871220078792971238414479, −3.95105490868582339229663793418, −3.35866792301398925150096351414, 3.35866792301398925150096351414, 3.95105490868582339229663793418, 5.03108871220078792971238414479, 5.51855702707420985771365031225, 6.52279276308513502587000590701, 6.76486336296373504490087508573, 8.577406316803923368617198473940, 8.848449434936333430085111719598, 9.864497437365040705722664129784, 10.00079602267096372848508341429, 11.32610889315071847199510580771, 11.62116892256875750743888082634, 12.83759326146348050290394206512, 12.98060071194356131373890504545, 13.25989587995379888651411950046, 13.52877102884076407184858132043, 14.68264660679604815517282447134, 14.93259322616907854553657847490, 15.48698718155727381100159608334, 16.25893957648634954297268744775

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