Properties

Degree 4
Conductor $ 47^{2} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s + 21-s + 2·25-s + 32-s + 34-s − 37-s − 42-s + 2·47-s − 2·50-s + 51-s − 53-s − 59-s − 61-s − 64-s − 71-s + 74-s − 2·75-s − 79-s + 4·83-s − 89-s − 2·94-s − 96-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s − 7-s + 14-s − 17-s + 21-s + 2·25-s + 32-s + 34-s − 37-s − 42-s + 2·47-s − 2·50-s + 51-s − 53-s − 59-s − 61-s − 64-s − 71-s + 74-s − 2·75-s − 79-s + 4·83-s − 89-s − 2·94-s − 96-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2209\)    =    \(47^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{47} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 2209,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.08912006073$
$L(\frac12)$  $\approx$  $0.08912006073$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 47$, \(F_p\) is a polynomial of degree 4. If $p = 47$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad47$C_1$ \( ( 1 - T )^{2} \)
good2$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
59$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
61$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
97$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
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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

−16.39761299926307357992794402644, −16.07886948847836249116832192876, −15.30703876774217792567430703967, −14.90531002592942505063566570766, −13.81816516142158278342030488727, −13.54757720598963822818329142040, −12.50277346124156882218469552755, −12.42821254608935038877218490071, −11.52817428198532142050518472672, −10.84649755632872285874832993509, −10.50369075022612179601131080455, −9.667695505824941703172707285159, −8.974116803878374901191307120128, −8.810223959306080153905100573024, −7.70850801210328648392120929641, −6.73380655474091416962152422615, −6.34108978922784854587300086860, −5.40411667131609441039217306830, −4.44331727518334815292959324093, −2.98743469578434504405692118290, 2.98743469578434504405692118290, 4.44331727518334815292959324093, 5.40411667131609441039217306830, 6.34108978922784854587300086860, 6.73380655474091416962152422615, 7.70850801210328648392120929641, 8.810223959306080153905100573024, 8.974116803878374901191307120128, 9.667695505824941703172707285159, 10.50369075022612179601131080455, 10.84649755632872285874832993509, 11.52817428198532142050518472672, 12.42821254608935038877218490071, 12.50277346124156882218469552755, 13.54757720598963822818329142040, 13.81816516142158278342030488727, 14.90531002592942505063566570766, 15.30703876774217792567430703967, 16.07886948847836249116832192876, 16.39761299926307357992794402644

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