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

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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  :  $\chi_{28900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 28900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.087828865$
$L(\frac12)$  $\approx$  $2.087828865$
$L(\frac{3}{2})$   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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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.71369463785094807460861887954, −9.673699245760630854857163727186, −9.595427966477503798189291150409, −8.777267495738734926545470281019, −8.553894901425455625758113621537, −8.051648952256609540644552125398, −7.73819026466125478177770246183, −6.40224127653083741685001868836, −6.12219164428167399966994400260, −5.48830190311268484103928206059, −4.87878126295597667599082294631, −4.14800988415631466917326004810, −3.16218883380341411967148272410, −2.09774505601165615279768778452, −2.01602626977355442406190385855, 2.01602626977355442406190385855, 2.09774505601165615279768778452, 3.16218883380341411967148272410, 4.14800988415631466917326004810, 4.87878126295597667599082294631, 5.48830190311268484103928206059, 6.12219164428167399966994400260, 6.40224127653083741685001868836, 7.73819026466125478177770246183, 8.051648952256609540644552125398, 8.553894901425455625758113621537, 8.777267495738734926545470281019, 9.595427966477503798189291150409, 9.673699245760630854857163727186, 10.71369463785094807460861887954

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