Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 67^{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 − 2·5-s + 2·6-s − 2·7-s − 8-s − 3·9-s − 2·10-s − 6·11-s − 2·13-s − 2·14-s − 4·15-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 4·21-s − 6·22-s + 6·23-s − 2·24-s + 3·25-s − 2·26-s − 14·27-s + 6·29-s − 4·30-s − 5·31-s − 12·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 0.894·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s − 9-s − 0.632·10-s − 1.80·11-s − 0.554·13-s − 0.534·14-s − 1.03·15-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.872·21-s − 1.27·22-s + 1.25·23-s − 0.408·24-s + 3/5·25-s − 0.392·26-s − 2.69·27-s + 1.11·29-s − 0.730·30-s − 0.898·31-s − 2.08·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 448900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 448900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(448900\)    =    \(2^{2} \cdot 5^{2} \cdot 67^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{670} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 448900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.15278$
$L(\frac12)$  $\approx$  $1.15278$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;67\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;67\}$, 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$C_1$ \( ( 1 + T )^{2} \)
67$C_2$ \( 1 - 5 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−10.98541347834296746446520843679, −10.44842362966500720226932090160, −9.618538959847183781721636774267, −9.272174820517103042001349727963, −9.032645143961108496014525540660, −8.635654732808729228666065182018, −7.937866989816557168551649806659, −7.66829807172261313648704933573, −7.60937830845796345154928176191, −6.54296788073615386345437574074, −6.45405413344596892238755058393, −5.44825903193512128852002821382, −5.36436203103784840998680545498, −4.68449811645055221549683136102, −4.21766230232826466539162546634, −3.31399820513476425328931648729, −3.23261278261940623949100268241, −2.48155940693625889045567456367, −2.46626714514537242760146275771, −0.44634355792084626236331203684, 0.44634355792084626236331203684, 2.46626714514537242760146275771, 2.48155940693625889045567456367, 3.23261278261940623949100268241, 3.31399820513476425328931648729, 4.21766230232826466539162546634, 4.68449811645055221549683136102, 5.36436203103784840998680545498, 5.44825903193512128852002821382, 6.45405413344596892238755058393, 6.54296788073615386345437574074, 7.60937830845796345154928176191, 7.66829807172261313648704933573, 7.937866989816557168551649806659, 8.635654732808729228666065182018, 9.032645143961108496014525540660, 9.272174820517103042001349727963, 9.618538959847183781721636774267, 10.44842362966500720226932090160, 10.98541347834296746446520843679

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