Properties

Label 2-4019-1.1-c1-0-71
Degree $2$
Conductor $4019$
Sign $1$
Analytic cond. $32.0918$
Root an. cond. $5.66496$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.60·2-s − 0.581·3-s + 4.77·4-s + 3.23·5-s + 1.51·6-s − 2.22·7-s − 7.22·8-s − 2.66·9-s − 8.40·10-s + 4.03·11-s − 2.77·12-s − 4.71·13-s + 5.79·14-s − 1.87·15-s + 9.24·16-s − 2.68·17-s + 6.92·18-s + 5.82·19-s + 15.4·20-s + 1.29·21-s − 10.4·22-s − 3.12·23-s + 4.20·24-s + 5.43·25-s + 12.2·26-s + 3.29·27-s − 10.6·28-s + ⋯
L(s)  = 1  − 1.84·2-s − 0.335·3-s + 2.38·4-s + 1.44·5-s + 0.618·6-s − 0.841·7-s − 2.55·8-s − 0.887·9-s − 2.65·10-s + 1.21·11-s − 0.801·12-s − 1.30·13-s + 1.54·14-s − 0.485·15-s + 2.31·16-s − 0.652·17-s + 1.63·18-s + 1.33·19-s + 3.44·20-s + 0.282·21-s − 2.23·22-s − 0.650·23-s + 0.857·24-s + 1.08·25-s + 2.40·26-s + 0.633·27-s − 2.00·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4019\)
Sign: $1$
Analytic conductor: \(32.0918\)
Root analytic conductor: \(5.66496\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4019,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6370881511\)
\(L(\frac12)\) \(\approx\) \(0.6370881511\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4019 \( 1+O(T) \)
good2 \( 1 + 2.60T + 2T^{2} \)
3 \( 1 + 0.581T + 3T^{2} \)
5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 2.22T + 7T^{2} \)
11 \( 1 - 4.03T + 11T^{2} \)
13 \( 1 + 4.71T + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 - 3.30T + 29T^{2} \)
31 \( 1 + 9.01T + 31T^{2} \)
37 \( 1 - 4.13T + 37T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 + 8.22T + 43T^{2} \)
47 \( 1 - 3.83T + 47T^{2} \)
53 \( 1 + 5.42T + 53T^{2} \)
59 \( 1 - 3.04T + 59T^{2} \)
61 \( 1 - 5.32T + 61T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 - 7.47T + 71T^{2} \)
73 \( 1 - 0.456T + 73T^{2} \)
79 \( 1 - 8.71T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.828792920975655400512831180747, −7.78512553304396377500236154651, −6.98612791071662479629519309069, −6.42015533814189998506356278858, −5.92006740273329889862524312363, −5.05023823162111616382391618388, −3.36145979469378182601401357125, −2.46997895595387086279613725691, −1.77867308477231643962060411847, −0.59794807919077894515599692099, 0.59794807919077894515599692099, 1.77867308477231643962060411847, 2.46997895595387086279613725691, 3.36145979469378182601401357125, 5.05023823162111616382391618388, 5.92006740273329889862524312363, 6.42015533814189998506356278858, 6.98612791071662479629519309069, 7.78512553304396377500236154651, 8.828792920975655400512831180747

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