Properties

Label 2-19-19.5-c1-0-0
Degree $2$
Conductor $19$
Sign $0.992 - 0.120i$
Analytic cond. $0.151715$
Root an. cond. $0.389507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 + 0.300i)2-s + (0.0923 − 0.524i)3-s + (−0.939 + 0.788i)4-s + (−1.93 − 1.62i)5-s + (0.0812 + 0.460i)6-s + (0.939 + 1.62i)7-s + (1.41 − 2.45i)8-s + (2.55 + 0.929i)9-s + (2.09 + 0.761i)10-s + (−1.70 + 2.95i)11-s + (0.326 + 0.565i)12-s + (−0.918 − 5.21i)13-s + (−1.26 − 1.06i)14-s + (−1.03 + 0.866i)15-s + (−0.00727 + 0.0412i)16-s + (−1.55 + 0.565i)17-s + ⋯
L(s)  = 1  + (−0.584 + 0.212i)2-s + (0.0533 − 0.302i)3-s + (−0.469 + 0.394i)4-s + (−0.867 − 0.727i)5-s + (0.0331 + 0.188i)6-s + (0.355 + 0.615i)7-s + (0.501 − 0.868i)8-s + (0.851 + 0.309i)9-s + (0.661 + 0.240i)10-s + (−0.514 + 0.890i)11-s + (0.0942 + 0.163i)12-s + (−0.254 − 1.44i)13-s + (−0.338 − 0.283i)14-s + (−0.266 + 0.223i)15-s + (−0.00181 + 0.0103i)16-s + (−0.376 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(0.151715\)
Root analytic conductor: \(0.389507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424941 + 0.0256138i\)
\(L(\frac12)\) \(\approx\) \(0.424941 + 0.0256138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (2.52 - 3.55i)T \)
good2 \( 1 + (0.826 - 0.300i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.0923 + 0.524i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (1.93 + 1.62i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.939 - 1.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.918 + 5.21i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.55 - 0.565i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-1.34 + 1.13i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.25 - 1.18i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.971 + 1.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.837T + 37T^{2} \)
41 \( 1 + (0.779 - 4.42i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.67 - 3.08i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (0.673 + 0.245i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.67 - 3.92i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.1 + 3.67i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.36 + 2.82i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (13.3 + 4.86i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (10.5 + 8.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.30 - 7.40i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.20 - 6.85i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.25 + 2.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.396 + 2.24i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.71 + 0.623i)T + (74.3 - 62.3i)T^{2} \)
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

−18.46918918689960095650841774407, −17.60499122914411020778979722704, −16.17732776717709056814226751993, −15.18410297518871349846975131815, −12.94507516780994903289960016036, −12.39450856827142164252646559524, −10.16429174854356309503426923748, −8.425980951481141939577225884869, −7.60731457672383664023175471594, −4.62462035195005422634167108817, 4.34529178975668977150684737643, 7.17829429165426303949443463935, 8.893438845980321839455126817252, 10.44393828019207959773495584508, 11.38287959847209645922429688582, 13.59338982527935148343927672618, 14.78702621559241815194078186146, 16.07899228041203536116307963144, 17.59806585637227498193811006301, 18.88860600195362025374912198094

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