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Properties

Label 2-19-19.12-c2-0-0
Degree $2$
Conductor $19$
Sign $-0.670 - 0.742i$
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  + (−2.90 + 1.67i)2-s + (−3.29 + 1.90i)3-s + (3.63 − 6.29i)4-s + (3.47 + 6.01i)5-s + (6.37 − 11.0i)6-s − 1.22·7-s + 10.9i·8-s + (2.72 − 4.72i)9-s + (−20.1 − 11.6i)10-s − 0.0363·11-s + 27.6i·12-s + (14.6 + 8.44i)13-s + (3.57 − 2.06i)14-s + (−22.8 − 13.2i)15-s + (−3.86 − 6.68i)16-s + (−4.59 − 7.95i)17-s + ⋯
L(s)  = 1  + (−1.45 + 0.839i)2-s + (−1.09 + 0.633i)3-s + (0.908 − 1.57i)4-s + (0.694 + 1.20i)5-s + (1.06 − 1.84i)6-s − 0.175·7-s + 1.36i·8-s + (0.302 − 0.524i)9-s + (−2.01 − 1.16i)10-s − 0.00330·11-s + 2.30i·12-s + (1.12 + 0.649i)13-s + (0.255 − 0.147i)14-s + (−1.52 − 0.880i)15-s + (−0.241 − 0.418i)16-s + (−0.270 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.670 - 0.742i$
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: \((2,\ 19,\ (\ :1),\ -0.670 - 0.742i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.146831 + 0.330521i\)
\(L(\frac12)\) \(\approx\) \(0.146831 + 0.330521i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-12.7 + 14.0i)T \)
good2 \( 1 + (2.90 - 1.67i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (3.29 - 1.90i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.47 - 6.01i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 1.22T + 49T^{2} \)
11 \( 1 + 0.0363T + 121T^{2} \)
13 \( 1 + (-14.6 - 8.44i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (4.59 + 7.95i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (4.87 - 8.45i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-4.50 - 2.60i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 44.3iT - 961T^{2} \)
37 \( 1 + 45.5iT - 1.36e3T^{2} \)
41 \( 1 + (-50.0 + 28.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-15.1 - 26.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-25.5 + 44.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-10.0 - 5.82i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (17.7 - 10.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (8.23 - 14.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (1.83 + 1.05i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (33.7 - 19.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (28.1 + 48.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-39.8 + 23.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 65.3T + 6.88e3T^{2} \)
89 \( 1 + (134. + 77.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (98.8 - 57.0i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−18.07586895489502464980063088476, −17.76649213455984656020385038920, −16.36611813417837925252495550651, −15.67969475235867680485329370342, −14.06231304064199745424727017450, −11.21028007711866386168629332847, −10.43493749144900231436596002593, −9.205692631029819668959370032924, −6.99085913654587462377482131703, −5.90275402424136128608612692557, 1.13742909390690919982331790996, 5.92418137942803284415178909158, 8.167788986407499217854790031268, 9.549447942268131998036406479746, 10.95734516537837958099481812246, 12.19228454183523195142567086700, 13.15026717554769433189233991030, 16.24591486189916247809123663715, 17.10976096016469762580637297253, 17.87360466346166401278990870296

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