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  − 2.69·2-s + 3-s + 5.26·4-s − 1.49·5-s − 2.69·6-s + 1.50·7-s − 8.78·8-s + 9-s + 4.01·10-s − 3.43·11-s + 5.26·12-s + 4.91·13-s − 4.06·14-s − 1.49·15-s + 13.1·16-s + 3.52·17-s − 2.69·18-s + 2.70·19-s − 7.83·20-s + 1.50·21-s + 9.25·22-s + 2.27·23-s − 8.78·24-s − 2.77·25-s − 13.2·26-s + 27-s + 7.92·28-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.577·3-s + 2.63·4-s − 0.666·5-s − 1.10·6-s + 0.569·7-s − 3.10·8-s + 0.333·9-s + 1.26·10-s − 1.03·11-s + 1.51·12-s + 1.36·13-s − 1.08·14-s − 0.384·15-s + 3.28·16-s + 0.856·17-s − 0.635·18-s + 0.619·19-s − 1.75·20-s + 0.328·21-s + 1.97·22-s + 0.475·23-s − 1.79·24-s − 0.555·25-s − 2.59·26-s + 0.192·27-s + 1.49·28-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  =  $(2,\ 8013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.091190174$
$L(\frac12)$  $\approx$  $1.091190174$
$L(\frac{3}{2})$   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 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 + 1.49T + 5T^{2} \)
7 \( 1 - 1.50T + 7T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 - 4.91T + 13T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 - 2.27T + 23T^{2} \)
29 \( 1 - 1.34T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
37 \( 1 - 2.96T + 37T^{2} \)
41 \( 1 - 6.75T + 41T^{2} \)
43 \( 1 - 4.68T + 43T^{2} \)
47 \( 1 + 4.45T + 47T^{2} \)
53 \( 1 - 2.19T + 53T^{2} \)
59 \( 1 - 7.53T + 59T^{2} \)
61 \( 1 - 0.353T + 61T^{2} \)
67 \( 1 + 0.363T + 67T^{2} \)
71 \( 1 - 5.88T + 71T^{2} \)
73 \( 1 + 1.33T + 73T^{2} \)
79 \( 1 - 4.59T + 79T^{2} \)
83 \( 1 + 9.77T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 7.45T + 97T^{2} \)
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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

−7.907804850272973752963149806212, −7.71922675408462763263899715893, −6.84037165938279791660189942816, −6.05255727055238547011435443718, −5.23096037468053339188644639389, −3.99484778239805179177611258035, −3.13823286481042839469566499417, −2.47661571847057596094389524546, −1.42498421956711669037774088059, −0.73215729879918039811298976443, 0.73215729879918039811298976443, 1.42498421956711669037774088059, 2.47661571847057596094389524546, 3.13823286481042839469566499417, 3.99484778239805179177611258035, 5.23096037468053339188644639389, 6.05255727055238547011435443718, 6.84037165938279791660189942816, 7.71922675408462763263899715893, 7.907804850272973752963149806212

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