Properties

Degree 2
Conductor $ 71 \cdot 113 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.43·2-s − 0.327·3-s + 3.94·4-s − 2.07·5-s + 0.798·6-s − 0.184·7-s − 4.73·8-s − 2.89·9-s + 5.06·10-s + 6.19·11-s − 1.29·12-s + 5.93·13-s + 0.448·14-s + 0.679·15-s + 3.65·16-s − 1.22·17-s + 7.05·18-s + 6.79·19-s − 8.18·20-s + 0.0602·21-s − 15.0·22-s + 4.78·23-s + 1.54·24-s − 0.689·25-s − 14.4·26-s + 1.93·27-s − 0.725·28-s + ⋯
L(s)  = 1  − 1.72·2-s − 0.189·3-s + 1.97·4-s − 0.928·5-s + 0.325·6-s − 0.0695·7-s − 1.67·8-s − 0.964·9-s + 1.60·10-s + 1.86·11-s − 0.372·12-s + 1.64·13-s + 0.119·14-s + 0.175·15-s + 0.912·16-s − 0.296·17-s + 1.66·18-s + 1.55·19-s − 1.82·20-s + 0.0131·21-s − 3.21·22-s + 0.998·23-s + 0.316·24-s − 0.137·25-s − 2.83·26-s + 0.371·27-s − 0.137·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8023\)    =    \(71 \cdot 113\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8023} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8023,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{71,\;113\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{71,\;113\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad71 \( 1 + T \)
113 \( 1 + T \)
good2 \( 1 + 2.43T + 2T^{2} \)
3 \( 1 + 0.327T + 3T^{2} \)
5 \( 1 + 2.07T + 5T^{2} \)
7 \( 1 + 0.184T + 7T^{2} \)
11 \( 1 - 6.19T + 11T^{2} \)
13 \( 1 - 5.93T + 13T^{2} \)
17 \( 1 + 1.22T + 17T^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
23 \( 1 - 4.78T + 23T^{2} \)
29 \( 1 + 1.27T + 29T^{2} \)
31 \( 1 + 3.43T + 31T^{2} \)
37 \( 1 - 6.19T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 - 0.000567T + 59T^{2} \)
61 \( 1 + 3.47T + 61T^{2} \)
67 \( 1 - 0.649T + 67T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 0.0342T + 89T^{2} \)
97 \( 1 + 4.90T + 97T^{2} \)
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

−7.75653812531627464583109640628, −6.82653569550628245319692733173, −6.53459468995851242373811386358, −5.73890070991785600373111223234, −4.62033866661036671741606100422, −3.44247049300367600373959747968, −3.25490925301354972465594932961, −1.62644941272174030077036617839, −1.10867354828650742235012657409, 0, 1.10867354828650742235012657409, 1.62644941272174030077036617839, 3.25490925301354972465594932961, 3.44247049300367600373959747968, 4.62033866661036671741606100422, 5.73890070991785600373111223234, 6.53459468995851242373811386358, 6.82653569550628245319692733173, 7.75653812531627464583109640628

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