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 − 1.39·3-s + 4-s + 3.35·5-s + 1.39·6-s − 0.849·7-s − 8-s − 1.06·9-s − 3.35·10-s − 2.14·11-s − 1.39·12-s − 0.777·13-s + 0.849·14-s − 4.67·15-s + 16-s + 4.54·17-s + 1.06·18-s − 19-s + 3.35·20-s + 1.18·21-s + 2.14·22-s − 6.97·23-s + 1.39·24-s + 6.27·25-s + 0.777·26-s + 5.65·27-s − 0.849·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.803·3-s + 0.5·4-s + 1.50·5-s + 0.568·6-s − 0.321·7-s − 0.353·8-s − 0.353·9-s − 1.06·10-s − 0.646·11-s − 0.401·12-s − 0.215·13-s + 0.227·14-s − 1.20·15-s + 0.250·16-s + 1.10·17-s + 0.250·18-s − 0.229·19-s + 0.750·20-s + 0.258·21-s + 0.457·22-s − 1.45·23-s + 0.284·24-s + 1.25·25-s + 0.152·26-s + 1.08·27-s − 0.160·28-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 \( 1 + 1.39T + 3T^{2} \)
5 \( 1 - 3.35T + 5T^{2} \)
7 \( 1 + 0.849T + 7T^{2} \)
11 \( 1 + 2.14T + 11T^{2} \)
13 \( 1 + 0.777T + 13T^{2} \)
17 \( 1 - 4.54T + 17T^{2} \)
23 \( 1 + 6.97T + 23T^{2} \)
29 \( 1 + 0.251T + 29T^{2} \)
31 \( 1 + 8.19T + 31T^{2} \)
37 \( 1 - 7.04T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
47 \( 1 + 4.32T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 3.78T + 59T^{2} \)
61 \( 1 - 7.33T + 61T^{2} \)
67 \( 1 + 6.16T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 3.21T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 3.27T + 89T^{2} \)
97 \( 1 + 7.35T + 97T^{2} \)
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

−7.53189595597461437096843081016, −6.64018413953080161416037416917, −5.98955500557488174591384441824, −5.65752452209887466664791946954, −5.09423167380090548367184163173, −3.84205990245348609668688398641, −2.72272219845041916872107778115, −2.17018501210388722075525343577, −1.13425936938956169697005354295, 0, 1.13425936938956169697005354295, 2.17018501210388722075525343577, 2.72272219845041916872107778115, 3.84205990245348609668688398641, 5.09423167380090548367184163173, 5.65752452209887466664791946954, 5.98955500557488174591384441824, 6.64018413953080161416037416917, 7.53189595597461437096843081016

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