Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
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-s − 2.52·3-s + 4-s − 1.30·5-s + 2.52·6-s + 0.799·7-s − 8-s + 3.35·9-s + 1.30·10-s − 2.77·11-s − 2.52·12-s + 3.89·13-s − 0.799·14-s + 3.29·15-s + 16-s + 7.65·17-s − 3.35·18-s − 19-s − 1.30·20-s − 2.01·21-s + 2.77·22-s − 0.364·23-s + 2.52·24-s − 3.29·25-s − 3.89·26-s − 0.902·27-s + 0.799·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.45·3-s + 0.5·4-s − 0.583·5-s + 1.02·6-s + 0.302·7-s − 0.353·8-s + 1.11·9-s + 0.412·10-s − 0.837·11-s − 0.727·12-s + 1.08·13-s − 0.213·14-s + 0.849·15-s + 0.250·16-s + 1.85·17-s − 0.791·18-s − 0.229·19-s − 0.291·20-s − 0.439·21-s + 0.592·22-s − 0.0759·23-s + 0.514·24-s − 0.659·25-s − 0.764·26-s − 0.173·27-s + 0.151·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
19 \( 1 + T \)
211 \( 1 + T \)
good3 \( 1 + 2.52T + 3T^{2} \)
5 \( 1 + 1.30T + 5T^{2} \)
7 \( 1 - 0.799T + 7T^{2} \)
11 \( 1 + 2.77T + 11T^{2} \)
13 \( 1 - 3.89T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
23 \( 1 + 0.364T + 23T^{2} \)
29 \( 1 - 1.06T + 29T^{2} \)
31 \( 1 + 6.08T + 31T^{2} \)
37 \( 1 + 0.605T + 37T^{2} \)
41 \( 1 + 6.71T + 41T^{2} \)
43 \( 1 + 8.72T + 43T^{2} \)
47 \( 1 - 4.28T + 47T^{2} \)
53 \( 1 - 4.28T + 53T^{2} \)
59 \( 1 - 3.47T + 59T^{2} \)
61 \( 1 - 0.0120T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 5.61T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 5.00T + 97T^{2} \)
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

−7.46424620364377761494563504419, −6.87526033487713369256812118229, −6.01307778728579091842891273604, −5.52694296410180698559417127941, −4.96881023153971304838682681540, −3.86779073930678690252207964538, −3.20118471165529689520266650302, −1.82004218632450635531285194376, −0.943095196483826179719305998639, 0, 0.943095196483826179719305998639, 1.82004218632450635531285194376, 3.20118471165529689520266650302, 3.86779073930678690252207964538, 4.96881023153971304838682681540, 5.52694296410180698559417127941, 6.01307778728579091842891273604, 6.87526033487713369256812118229, 7.46424620364377761494563504419

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