Properties

Degree 2
Conductor 47
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s − 1.61·3-s − 0.618·4-s − 1.00·6-s + 0.618·7-s − 8-s + 1.61·9-s + 0.999·12-s + 0.381·14-s − 1.61·17-s + 1.00·18-s − 1.00·21-s + 1.61·24-s + 25-s − 27-s − 0.381·28-s + 0.999·32-s − 1.00·34-s − 0.999·36-s − 1.61·37-s − 0.618·42-s + 47-s − 0.618·49-s + 0.618·50-s + 2.61·51-s + 0.618·53-s − 0.618·54-s + ⋯
L(s)  = 1  + 0.618·2-s − 1.61·3-s − 0.618·4-s − 1.00·6-s + 0.618·7-s − 8-s + 1.61·9-s + 0.999·12-s + 0.381·14-s − 1.61·17-s + 1.00·18-s − 1.00·21-s + 1.61·24-s + 25-s − 27-s − 0.381·28-s + 0.999·32-s − 1.00·34-s − 0.999·36-s − 1.61·37-s − 0.618·42-s + 47-s − 0.618·49-s + 0.618·50-s + 2.61·51-s + 0.618·53-s − 0.618·54-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{47} (46, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 47,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.3597283851$
$L(\frac12)$  $\approx$  $0.3597283851$
$L(1)$   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 2. If $p = 47$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad47 \( 1 - T \)
good2 \( 1 - 0.618T + T^{2} \)
3 \( 1 + 1.61T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.61T + T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.07886948847836249116832192876, −14.90531002592942505063566570766, −13.54757720598963822818329142040, −12.50277346124156882218469552755, −11.52817428198532142050518472672, −10.50369075022612179601131080455, −8.810223959306080153905100573024, −6.73380655474091416962152422615, −5.40411667131609441039217306830, −4.44331727518334815292959324093, 4.44331727518334815292959324093, 5.40411667131609441039217306830, 6.73380655474091416962152422615, 8.810223959306080153905100573024, 10.50369075022612179601131080455, 11.52817428198532142050518472672, 12.50277346124156882218469552755, 13.54757720598963822818329142040, 14.90531002592942505063566570766, 16.07886948847836249116832192876

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