Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 17^{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  − 2·3-s − 4-s − 4·5-s − 4·7-s − 3·9-s + 2·12-s + 8·15-s + 16-s − 2·17-s − 10·19-s + 4·20-s + 8·21-s + 8·23-s + 11·25-s + 14·27-s + 4·28-s + 16·35-s + 3·36-s − 4·37-s + 12·45-s − 2·48-s − 2·49-s + 4·51-s + 20·57-s − 10·59-s − 8·60-s + 12·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 1.78·5-s − 1.51·7-s − 9-s + 0.577·12-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 2.29·19-s + 0.894·20-s + 1.74·21-s + 1.66·23-s + 11/5·25-s + 2.69·27-s + 0.755·28-s + 2.70·35-s + 1/2·36-s − 0.657·37-s + 1.78·45-s − 0.288·48-s − 2/7·49-s + 0.560·51-s + 2.64·57-s − 1.30·59-s − 1.03·60-s + 1.51·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(28900\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{170} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 28900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.0835131$
$L(\frac12)$  $\approx$  $0.0835131$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−13.19512286103300492115631524185, −12.39050079814818103875850422595, −12.15195208048770882825759031227, −11.35065363829323746908868029367, −11.25689181617923631173042783422, −10.52435747411357564780555327956, −10.48558930738402783262927176327, −9.225353027840115778268430985145, −9.027226062417982679320643995668, −8.365504559433248474709823801345, −8.102116035085813470266632876031, −6.93565938255421960397609114452, −6.79457098757508507053371387289, −6.15832968561240465992565053715, −5.53796867274967089762237965035, −4.67218221910896069750003341173, −4.31829051465682132741427742833, −3.29356522318068060642161603631, −2.92968787802752899023078845282, −0.27902629164556144119710329257, 0.27902629164556144119710329257, 2.92968787802752899023078845282, 3.29356522318068060642161603631, 4.31829051465682132741427742833, 4.67218221910896069750003341173, 5.53796867274967089762237965035, 6.15832968561240465992565053715, 6.79457098757508507053371387289, 6.93565938255421960397609114452, 8.102116035085813470266632876031, 8.365504559433248474709823801345, 9.027226062417982679320643995668, 9.225353027840115778268430985145, 10.48558930738402783262927176327, 10.52435747411357564780555327956, 11.25689181617923631173042783422, 11.35065363829323746908868029367, 12.15195208048770882825759031227, 12.39050079814818103875850422595, 13.19512286103300492115631524185

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