Properties

Degree 2
Conductor $ 13 \cdot 619 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 0.381·3-s − 1.61·4-s + 3.23·5-s + 0.236·6-s − 2.61·7-s + 2.23·8-s − 2.85·9-s − 2.00·10-s + 5·11-s + 0.618·12-s − 13-s + 1.61·14-s − 1.23·15-s + 1.85·16-s + 4.23·17-s + 1.76·18-s − 8.23·19-s − 5.23·20-s + 21-s − 3.09·22-s − 4·23-s − 0.854·24-s + 5.47·25-s + 0.618·26-s + 2.23·27-s + 4.23·28-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.220·3-s − 0.809·4-s + 1.44·5-s + 0.0963·6-s − 0.989·7-s + 0.790·8-s − 0.951·9-s − 0.632·10-s + 1.50·11-s + 0.178·12-s − 0.277·13-s + 0.432·14-s − 0.319·15-s + 0.463·16-s + 1.02·17-s + 0.415·18-s − 1.88·19-s − 1.17·20-s + 0.218·21-s − 0.658·22-s − 0.834·23-s − 0.174·24-s + 1.09·25-s + 0.121·26-s + 0.430·27-s + 0.800·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8047\)    =    \(13 \cdot 619\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8047} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8047,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.149575433$
$L(\frac12)$  $\approx$  $1.149575433$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;619\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;619\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 \( 1 + T \)
619 \( 1 - T \)
good2 \( 1 + 0.618T + 2T^{2} \)
3 \( 1 + 0.381T + 3T^{2} \)
5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 + 8.23T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 7.23T + 29T^{2} \)
31 \( 1 + 6.23T + 31T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 - 4.23T + 41T^{2} \)
43 \( 1 + 9.23T + 43T^{2} \)
47 \( 1 - 5.14T + 47T^{2} \)
53 \( 1 - 5.70T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 - 9.18T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + 9.47T + 79T^{2} \)
83 \( 1 + 1.76T + 83T^{2} \)
89 \( 1 - 0.145T + 89T^{2} \)
97 \( 1 - 3.90T + 97T^{2} \)
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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

−8.160808142820660269042076563662, −6.78539019149311761263797141003, −6.53047568122544348573481725286, −5.72446600462784541316446902832, −5.32172201460189161230890811826, −4.20241811247347596179618538660, −3.59427422183073332827817802558, −2.51826035598364231014167048931, −1.65837434190666035108952209561, −0.58497649156770714201763802710, 0.58497649156770714201763802710, 1.65837434190666035108952209561, 2.51826035598364231014167048931, 3.59427422183073332827817802558, 4.20241811247347596179618538660, 5.32172201460189161230890811826, 5.72446600462784541316446902832, 6.53047568122544348573481725286, 6.78539019149311761263797141003, 8.160808142820660269042076563662

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