Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.58·2-s + 3-s + 4.70·4-s − 3.13·5-s − 2.58·6-s + 5.18·7-s − 6.99·8-s + 9-s + 8.12·10-s + 3.07·11-s + 4.70·12-s + 1.01·13-s − 13.4·14-s − 3.13·15-s + 8.71·16-s + 7.25·17-s − 2.58·18-s + 5.96·19-s − 14.7·20-s + 5.18·21-s − 7.96·22-s − 7.91·23-s − 6.99·24-s + 4.83·25-s − 2.62·26-s + 27-s + 24.3·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.577·3-s + 2.35·4-s − 1.40·5-s − 1.05·6-s + 1.95·7-s − 2.47·8-s + 0.333·9-s + 2.56·10-s + 0.928·11-s + 1.35·12-s + 0.280·13-s − 3.58·14-s − 0.809·15-s + 2.17·16-s + 1.75·17-s − 0.610·18-s + 1.36·19-s − 3.29·20-s + 1.13·21-s − 1.69·22-s − 1.65·23-s − 1.42·24-s + 0.967·25-s − 0.514·26-s + 0.192·27-s + 4.60·28-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  =  $(2,\ 8013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.264408884$
$L(\frac12)$  $\approx$  $1.264408884$
$L(\frac{3}{2})$   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 \( 1 + 2.58T + 2T^{2} \)
5 \( 1 + 3.13T + 5T^{2} \)
7 \( 1 - 5.18T + 7T^{2} \)
11 \( 1 - 3.07T + 11T^{2} \)
13 \( 1 - 1.01T + 13T^{2} \)
17 \( 1 - 7.25T + 17T^{2} \)
19 \( 1 - 5.96T + 19T^{2} \)
23 \( 1 + 7.91T + 23T^{2} \)
29 \( 1 + 8.81T + 29T^{2} \)
31 \( 1 + 6.72T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 + 2.30T + 41T^{2} \)
43 \( 1 - 1.33T + 43T^{2} \)
47 \( 1 - 5.80T + 47T^{2} \)
53 \( 1 - 2.49T + 53T^{2} \)
59 \( 1 - 6.15T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + 7.41T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 0.487T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 2.15T + 89T^{2} \)
97 \( 1 - 1.02T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.83151250600807001800141781993, −7.51029570922081779132195020345, −7.25530012930608761915550967839, −5.89102644756264882642286650790, −5.13121374644737449158029392715, −3.79756095150980963512086904758, −3.64413461825135707412888037183, −2.17443526074381585425852989444, −1.49465406485753835510729937498, −0.791145769654338539607590409336, 0.791145769654338539607590409336, 1.49465406485753835510729937498, 2.17443526074381585425852989444, 3.64413461825135707412888037183, 3.79756095150980963512086904758, 5.13121374644737449158029392715, 5.89102644756264882642286650790, 7.25530012930608761915550967839, 7.51029570922081779132195020345, 7.83151250600807001800141781993

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