Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 3·9-s + 2·11-s − 4·13-s + 16-s − 2·17-s − 3·18-s − 2·19-s + 2·22-s + 4·23-s − 5·25-s − 4·26-s + 4·29-s + 4·31-s + 32-s − 2·34-s − 3·36-s + 2·37-s − 2·38-s + 6·41-s + 6·43-s + 2·44-s + 4·46-s − 47-s − 7·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 0.603·11-s − 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.458·19-s + 0.426·22-s + 0.834·23-s − 25-s − 0.784·26-s + 0.742·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.328·37-s − 0.324·38-s + 0.937·41-s + 0.914·43-s + 0.301·44-s + 0.589·46-s − 0.145·47-s − 49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 94 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 94 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(94\)    =    \(2 \cdot 47\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{94} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 94,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.356734506$
$L(\frac12)$  $\approx$  $1.356734506$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
47 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.69801811717119, −19.34389103297082, −17.59383111602883, −17.05975440894998, −15.86024439636576, −14.75977410872436, −14.16366362162334, −12.99281557844705, −11.93473672654547, −11.13582332876690, −9.733458342931033, −8.414023407790191, −6.987623487387878, −5.763396230983144, −4.395922429623901, −2.671820917250344, 2.671820917250344, 4.395922429623901, 5.763396230983144, 6.987623487387878, 8.414023407790191, 9.733458342931033, 11.13582332876690, 11.93473672654547, 12.99281557844705, 14.16366362162334, 14.75977410872436, 15.86024439636576, 17.05975440894998, 17.59383111602883, 19.34389103297082, 19.69801811717119

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