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Properties

Label 2-19-19.12-c2-0-2
Degree $2$
Conductor $19$
Sign $0.961 + 0.274i$
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  + (1.99 − 1.14i)2-s + (−3.70 + 2.13i)3-s + (0.640 − 1.10i)4-s + (−2.91 − 5.05i)5-s + (−4.91 + 8.50i)6-s + 9.38·7-s + 6.24i·8-s + (4.63 − 8.02i)9-s + (−11.6 − 6.71i)10-s − 4.66·11-s + 5.47i·12-s + (−4.96 − 2.86i)13-s + (18.6 − 10.7i)14-s + (21.6 + 12.4i)15-s + (9.74 + 16.8i)16-s + (−9.49 − 16.4i)17-s + ⋯
L(s)  = 1  + (0.995 − 0.574i)2-s + (−1.23 + 0.712i)3-s + (0.160 − 0.277i)4-s + (−0.583 − 1.01i)5-s + (−0.818 + 1.41i)6-s + 1.34·7-s + 0.781i·8-s + (0.514 − 0.891i)9-s + (−1.16 − 0.671i)10-s − 0.423·11-s + 0.456i·12-s + (−0.381 − 0.220i)13-s + (1.33 − 0.769i)14-s + (1.44 + 0.831i)15-s + (0.608 + 1.05i)16-s + (−0.558 − 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.961 + 0.274i$
Analytic conductor: \(0.517712\)
Root analytic conductor: \(0.719522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :1),\ 0.961 + 0.274i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.937331 - 0.131048i\)
\(L(\frac12)\) \(\approx\) \(0.937331 - 0.131048i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (3.40 - 18.6i)T \)
good2 \( 1 + (-1.99 + 1.14i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (3.70 - 2.13i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (2.91 + 5.05i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 9.38T + 49T^{2} \)
11 \( 1 + 4.66T + 121T^{2} \)
13 \( 1 + (4.96 + 2.86i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (9.49 + 16.4i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-6.41 + 11.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-27.7 - 16.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 26.9iT - 961T^{2} \)
37 \( 1 - 0.140iT - 1.36e3T^{2} \)
41 \( 1 + (-10.0 + 5.82i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (16.3 + 28.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (5.37 - 9.30i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (30.7 + 17.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (76.1 - 43.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-45.7 + 79.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.3 - 10.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (15.8 - 9.14i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-28.7 - 49.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (81.7 - 47.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 81.9T + 6.88e3T^{2} \)
89 \( 1 + (-17.3 - 10.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-117. + 67.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−17.92880955100365742881905003859, −16.93301446375177490428219687365, −15.78387818734613406689429622641, −14.26218322067127886099829670595, −12.51078823155909720358101421205, −11.73020812158164024945069527755, −10.71476085646487899152510447752, −8.317286874476701056503047280101, −5.15779616186179174069314923219, −4.57749720574230799147329532250, 4.77357460935657086079466821249, 6.33304559245663433956941880392, 7.52892446315695115894033998545, 10.82823240938914621247752772732, 11.74121906508611467267455215420, 13.18270717677479768273629001599, 14.61564884122049346344017253087, 15.47023759318547032022257150720, 17.23770555245501602632259664816, 18.17645069165681164424062427327

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