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 + 1.92·3-s + 4-s − 0.949·5-s − 1.92·6-s − 5.12·7-s − 8-s + 0.724·9-s + 0.949·10-s + 5.68·11-s + 1.92·12-s + 1.19·13-s + 5.12·14-s − 1.83·15-s + 16-s + 1.38·17-s − 0.724·18-s − 19-s − 0.949·20-s − 9.88·21-s − 5.68·22-s − 0.766·23-s − 1.92·24-s − 4.09·25-s − 1.19·26-s − 4.39·27-s − 5.12·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.11·3-s + 0.5·4-s − 0.424·5-s − 0.787·6-s − 1.93·7-s − 0.353·8-s + 0.241·9-s + 0.300·10-s + 1.71·11-s + 0.557·12-s + 0.330·13-s + 1.36·14-s − 0.473·15-s + 0.250·16-s + 0.335·17-s − 0.170·18-s − 0.229·19-s − 0.212·20-s − 2.15·21-s − 1.21·22-s − 0.159·23-s − 0.393·24-s − 0.819·25-s − 0.233·26-s − 0.845·27-s − 0.967·28-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 \( 1 - 1.92T + 3T^{2} \)
5 \( 1 + 0.949T + 5T^{2} \)
7 \( 1 + 5.12T + 7T^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
17 \( 1 - 1.38T + 17T^{2} \)
23 \( 1 + 0.766T + 23T^{2} \)
29 \( 1 - 1.61T + 29T^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 - 4.06T + 37T^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 8.05T + 47T^{2} \)
53 \( 1 + 3.78T + 53T^{2} \)
59 \( 1 - 3.49T + 59T^{2} \)
61 \( 1 + 6.12T + 61T^{2} \)
67 \( 1 - 9.03T + 67T^{2} \)
71 \( 1 + 1.73T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 - 4.84T + 79T^{2} \)
83 \( 1 + 0.862T + 83T^{2} \)
89 \( 1 - 4.83T + 89T^{2} \)
97 \( 1 + 1.09T + 97T^{2} \)
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

−7.59264000450247136296173139211, −6.89042713503851427857872153313, −6.30399709332378243094759608280, −5.80688854561255363284717063483, −4.16103738878601922862095021801, −3.68310804799235638580018438321, −3.13305153569889748827134896498, −2.34034139152775178829200697550, −1.20964717286042176384681077076, 0, 1.20964717286042176384681077076, 2.34034139152775178829200697550, 3.13305153569889748827134896498, 3.68310804799235638580018438321, 4.16103738878601922862095021801, 5.80688854561255363284717063483, 6.30399709332378243094759608280, 6.89042713503851427857872153313, 7.59264000450247136296173139211

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