Properties

Degree 2
Conductor $ 3 \cdot 2671 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.21·2-s + 3-s + 2.88·4-s + 1.51·5-s − 2.21·6-s + 0.0486·7-s − 1.96·8-s + 9-s − 3.35·10-s + 6.56·11-s + 2.88·12-s + 0.517·13-s − 0.107·14-s + 1.51·15-s − 1.43·16-s − 4.76·17-s − 2.21·18-s − 0.00792·19-s + 4.38·20-s + 0.0486·21-s − 14.5·22-s + 4.25·23-s − 1.96·24-s − 2.69·25-s − 1.14·26-s + 27-s + 0.140·28-s + ⋯
L(s)  = 1  − 1.56·2-s + 0.577·3-s + 1.44·4-s + 0.678·5-s − 0.902·6-s + 0.0183·7-s − 0.694·8-s + 0.333·9-s − 1.06·10-s + 1.97·11-s + 0.833·12-s + 0.143·13-s − 0.0287·14-s + 0.391·15-s − 0.358·16-s − 1.15·17-s − 0.521·18-s − 0.00181·19-s + 0.980·20-s + 0.0106·21-s − 3.09·22-s + 0.886·23-s − 0.400·24-s − 0.539·25-s − 0.224·26-s + 0.192·27-s + 0.0265·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8013\)    =    \(3 \cdot 2671\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8013} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.481500148$
$L(\frac12)$  $\approx$  $1.481500148$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;2671\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;2671\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
2671 \( 1 + T \)
good2 \( 1 + 2.21T + 2T^{2} \)
5 \( 1 - 1.51T + 5T^{2} \)
7 \( 1 - 0.0486T + 7T^{2} \)
11 \( 1 - 6.56T + 11T^{2} \)
13 \( 1 - 0.517T + 13T^{2} \)
17 \( 1 + 4.76T + 17T^{2} \)
19 \( 1 + 0.00792T + 19T^{2} \)
23 \( 1 - 4.25T + 23T^{2} \)
29 \( 1 + 7.88T + 29T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 - 7.82T + 37T^{2} \)
41 \( 1 + 7.47T + 41T^{2} \)
43 \( 1 + 8.77T + 43T^{2} \)
47 \( 1 - 1.80T + 47T^{2} \)
53 \( 1 - 5.72T + 53T^{2} \)
59 \( 1 - 0.207T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 8.59T + 67T^{2} \)
71 \( 1 - 0.245T + 71T^{2} \)
73 \( 1 - 9.82T + 73T^{2} \)
79 \( 1 - 16.6T + 79T^{2} \)
83 \( 1 - 6.64T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−8.037806271743441072809272609713, −7.24518350625726560585123537734, −6.63992479047448799248453519705, −6.25016334903479332214579519589, −5.06239455500280014607348833419, −4.09842770902792733251762627442, −3.37992564688319399506198755386, −2.05203371258239911885092002052, −1.80493551692842253308200127901, −0.76410445625170405408788654529, 0.76410445625170405408788654529, 1.80493551692842253308200127901, 2.05203371258239911885092002052, 3.37992564688319399506198755386, 4.09842770902792733251762627442, 5.06239455500280014607348833419, 6.25016334903479332214579519589, 6.63992479047448799248453519705, 7.24518350625726560585123537734, 8.037806271743441072809272609713

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