Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 3·5-s + 9-s − 3·11-s + 16-s − 6·19-s − 3·20-s + 4·25-s − 3·29-s + 10·31-s + 36-s + 12·41-s − 3·44-s − 3·45-s − 4·49-s + 9·55-s − 6·59-s + 16·61-s + 64-s − 9·71-s − 6·76-s − 2·79-s − 3·80-s − 8·81-s + 9·89-s + 18·95-s − 3·99-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.34·5-s + 1/3·9-s − 0.904·11-s + 1/4·16-s − 1.37·19-s − 0.670·20-s + 4/5·25-s − 0.557·29-s + 1.79·31-s + 1/6·36-s + 1.87·41-s − 0.452·44-s − 0.447·45-s − 4/7·49-s + 1.21·55-s − 0.781·59-s + 2.04·61-s + 1/8·64-s − 1.06·71-s − 0.688·76-s − 0.225·79-s − 0.335·80-s − 8/9·81-s + 0.953·89-s + 1.84·95-s − 0.301·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1900} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1900,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5856958784$
$L(\frac12)$  $\approx$  $0.5856958784$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 7 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
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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

−13.12629201823161730215632551035, −12.77978038682690811603907751033, −12.12462061109148388225220357108, −11.51959044376157204820634393232, −10.96581222542300546543217413733, −10.43360058383073541051440337968, −9.701139036768523738942555924982, −8.643755561385292296510754105134, −8.068899168563034845611018261769, −7.54335577564449427577768867852, −6.77565335418431214450302625944, −5.91364375610394068414214605262, −4.70053508669332129152077478238, −3.95180666919658488912421076947, −2.65340032178954426050786831781, 2.65340032178954426050786831781, 3.95180666919658488912421076947, 4.70053508669332129152077478238, 5.91364375610394068414214605262, 6.77565335418431214450302625944, 7.54335577564449427577768867852, 8.068899168563034845611018261769, 8.643755561385292296510754105134, 9.701139036768523738942555924982, 10.43360058383073541051440337968, 10.96581222542300546543217413733, 11.51959044376157204820634393232, 12.12462061109148388225220357108, 12.77978038682690811603907751033, 13.12629201823161730215632551035

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