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  − 1.61·2-s + 0.618·3-s + 1.61·4-s − 1.00·6-s − 1.61·7-s − 8-s − 0.618·9-s + 1.00·12-s + 2.61·14-s + 0.618·17-s + 0.999·18-s − 1.00·21-s − 0.618·24-s + 25-s − 27-s − 2.61·28-s + 32-s − 1.00·34-s − 0.999·36-s + 0.618·37-s + 1.61·42-s + 47-s + 1.61·49-s − 1.61·50-s + 0.381·51-s − 1.61·53-s + 1.61·54-s + ⋯
L(s)  = 1  − 1.61·2-s + 0.618·3-s + 1.61·4-s − 1.00·6-s − 1.61·7-s − 8-s − 0.618·9-s + 1.00·12-s + 2.61·14-s + 0.618·17-s + 0.999·18-s − 1.00·21-s − 0.618·24-s + 25-s − 27-s − 2.61·28-s + 32-s − 1.00·34-s − 0.999·36-s + 0.618·37-s + 1.61·42-s + 47-s + 1.61·49-s − 1.61·50-s + 0.381·51-s − 1.61·53-s + 1.61·54-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  =  $(2,\ 47,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.2477426425$
$L(\frac12)$  $\approx$  $0.2477426425$
$L(1)$   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 \( 1 + 1.61T + T^{2} \)
3 \( 1 - 0.618T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 0.618T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 0.618T + T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−16.39761299926307357992794402644, −15.30703876774217792567430703967, −13.81816516142158278342030488727, −12.42821254608935038877218490071, −10.84649755632872285874832993509, −9.667695505824941703172707285159, −8.974116803878374901191307120128, −7.70850801210328648392120929641, −6.34108978922784854587300086860, −2.98743469578434504405692118290, 2.98743469578434504405692118290, 6.34108978922784854587300086860, 7.70850801210328648392120929641, 8.974116803878374901191307120128, 9.667695505824941703172707285159, 10.84649755632872285874832993509, 12.42821254608935038877218490071, 13.81816516142158278342030488727, 15.30703876774217792567430703967, 16.39761299926307357992794402644

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