Properties

Degree 2
Conductor $ 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  − 2.70·2-s + 1.93·3-s + 5.30·4-s + 1.68·5-s − 5.24·6-s + 4.83·7-s − 8.94·8-s + 0.760·9-s − 4.56·10-s − 3.10·11-s + 10.2·12-s − 3.06·13-s − 13.0·14-s + 3.27·15-s + 13.5·16-s − 6.37·17-s − 2.05·18-s + 19-s + 8.96·20-s + 9.36·21-s + 8.40·22-s − 5.89·23-s − 17.3·24-s − 2.14·25-s + 8.29·26-s − 4.34·27-s + 25.6·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 1.11·3-s + 2.65·4-s + 0.755·5-s − 2.14·6-s + 1.82·7-s − 3.16·8-s + 0.253·9-s − 1.44·10-s − 0.937·11-s + 2.97·12-s − 0.851·13-s − 3.49·14-s + 0.845·15-s + 3.39·16-s − 1.54·17-s − 0.484·18-s + 0.229·19-s + 2.00·20-s + 2.04·21-s + 1.79·22-s − 1.22·23-s − 3.54·24-s − 0.429·25-s + 1.62·26-s − 0.835·27-s + 4.84·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4009\)    =    \(19 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4009} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 4009,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{19,\;211\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;211\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad19 \( 1 - T \)
211 \( 1 - T \)
good2 \( 1 + 2.70T + 2T^{2} \)
3 \( 1 - 1.93T + 3T^{2} \)
5 \( 1 - 1.68T + 5T^{2} \)
7 \( 1 - 4.83T + 7T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 + 3.06T + 13T^{2} \)
17 \( 1 + 6.37T + 17T^{2} \)
23 \( 1 + 5.89T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 + 7.88T + 31T^{2} \)
37 \( 1 - 6.79T + 37T^{2} \)
41 \( 1 - 6.98T + 41T^{2} \)
43 \( 1 + 6.77T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 9.18T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 - 5.04T + 61T^{2} \)
67 \( 1 + 3.69T + 67T^{2} \)
71 \( 1 - 0.0274T + 71T^{2} \)
73 \( 1 - 1.57T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 7.74T + 83T^{2} \)
89 \( 1 - 2.26T + 89T^{2} \)
97 \( 1 - 0.814T + 97T^{2} \)
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

−8.110149093701396317838159275602, −7.63682092538167261281739428762, −7.34602171481108484349276496539, −6.00592304333528233202065079923, −5.35549016686378117188073313948, −4.10123202754973698535664302029, −2.62492816769787744696622645754, −2.07582177394931746816831868855, −1.76851215338078852527621466493, 0, 1.76851215338078852527621466493, 2.07582177394931746816831868855, 2.62492816769787744696622645754, 4.10123202754973698535664302029, 5.35549016686378117188073313948, 6.00592304333528233202065079923, 7.34602171481108484349276496539, 7.63682092538167261281739428762, 8.110149093701396317838159275602

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