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Properties

Label 2-19-19.8-c2-0-0
Degree $2$
Conductor $19$
Sign $0.997 + 0.0666i$
Analytic cond. $0.517712$
Root an. cond. $0.719522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.583 − 0.336i)2-s + (2.49 + 1.43i)3-s + (−1.77 − 3.07i)4-s + (−1.55 + 2.69i)5-s + (−0.969 − 1.67i)6-s − 8.15·7-s + 5.08i·8-s + (−0.356 − 0.617i)9-s + (1.81 − 1.04i)10-s + 17.6·11-s − 10.2i·12-s + (5.33 − 3.07i)13-s + (4.75 + 2.74i)14-s + (−7.74 + 4.47i)15-s + (−5.37 + 9.31i)16-s + (−6.91 + 11.9i)17-s + ⋯
L(s)  = 1  + (−0.291 − 0.168i)2-s + (0.831 + 0.479i)3-s + (−0.443 − 0.767i)4-s + (−0.310 + 0.538i)5-s + (−0.161 − 0.279i)6-s − 1.16·7-s + 0.635i·8-s + (−0.0396 − 0.0686i)9-s + (0.181 − 0.104i)10-s + 1.60·11-s − 0.850i·12-s + (0.410 − 0.236i)13-s + (0.339 + 0.196i)14-s + (−0.516 + 0.298i)15-s + (−0.336 + 0.582i)16-s + (−0.406 + 0.704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.997 + 0.0666i$
Analytic conductor: \(0.517712\)
Root analytic conductor: \(0.719522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :1),\ 0.997 + 0.0666i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.796748 - 0.0265942i\)
\(L(\frac12)\) \(\approx\) \(0.796748 - 0.0265942i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-3.11 + 18.7i)T \)
good2 \( 1 + (0.583 + 0.336i)T + (2 + 3.46i)T^{2} \)
3 \( 1 + (-2.49 - 1.43i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.55 - 2.69i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 8.15T + 49T^{2} \)
11 \( 1 - 17.6T + 121T^{2} \)
13 \( 1 + (-5.33 + 3.07i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (6.91 - 11.9i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-2.46 - 4.27i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (38.2 - 22.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 43.0iT - 961T^{2} \)
37 \( 1 - 30.7iT - 1.36e3T^{2} \)
41 \( 1 + (28.6 + 16.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (15.7 - 27.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-8.86 - 15.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-14.6 + 8.45i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-20.3 - 11.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.52 + 14.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-84.9 + 49.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (1.36 + 0.786i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-2.85 + 4.94i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-41.9 - 24.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 74.2T + 6.88e3T^{2} \)
89 \( 1 + (-81.0 + 46.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (8.15 + 4.70i)T + (4.70e3 + 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.72569684968698391001815821541, −17.13233143300027670756543077017, −15.35530004276431807709925694410, −14.66693829321250198226123459291, −13.34016040523132652586436538275, −11.24548478149454533325174997406, −9.668510674184599677563624161801, −8.921182567136291675618346232195, −6.44748239747141298943803001821, −3.65326289968477166989528342053, 3.68820208513988294988307004110, 6.93697368223925667823325338341, 8.508360898295052111166777958608, 9.378271700993899991111338823306, 12.02795681264791752152545814693, 13.11765048719920882195238693115, 14.19620150917626694051639890287, 16.15605381997065699262481805020, 16.89560693464195141163938483488, 18.54821528792938220261238998213

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