Properties

Label 2-19-19.12-c4-0-3
Degree $2$
Conductor $19$
Sign $0.846 + 0.532i$
Analytic cond. $1.96402$
Root an. cond. $1.40143$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.52 − 1.45i)2-s + (8.18 − 4.72i)3-s + (−3.75 + 6.49i)4-s + (−7.00 − 12.1i)5-s + (13.7 − 23.8i)6-s + 19.8·7-s + 68.5i·8-s + (4.16 − 7.20i)9-s + (−35.3 − 20.4i)10-s − 177.·11-s + 70.8i·12-s + (83.2 + 48.0i)13-s + (50.0 − 28.8i)14-s + (−114. − 66.1i)15-s + (39.8 + 69.0i)16-s + (26.8 + 46.5i)17-s + ⋯
L(s)  = 1  + (0.631 − 0.364i)2-s + (0.909 − 0.525i)3-s + (−0.234 + 0.406i)4-s + (−0.280 − 0.484i)5-s + (0.382 − 0.662i)6-s + 0.404·7-s + 1.07i·8-s + (0.0513 − 0.0889i)9-s + (−0.353 − 0.204i)10-s − 1.46·11-s + 0.492i·12-s + (0.492 + 0.284i)13-s + (0.255 − 0.147i)14-s + (−0.509 − 0.294i)15-s + (0.155 + 0.269i)16-s + (0.0929 + 0.161i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.846 + 0.532i$
Analytic conductor: \(1.96402\)
Root analytic conductor: \(1.40143\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :2),\ 0.846 + 0.532i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.76963 - 0.510545i\)
\(L(\frac12)\) \(\approx\) \(1.76963 - 0.510545i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-161. + 322. i)T \)
good2 \( 1 + (-2.52 + 1.45i)T + (8 - 13.8i)T^{2} \)
3 \( 1 + (-8.18 + 4.72i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (7.00 + 12.1i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 19.8T + 2.40e3T^{2} \)
11 \( 1 + 177.T + 1.46e4T^{2} \)
13 \( 1 + (-83.2 - 48.0i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-26.8 - 46.5i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-372. + 644. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (738. + 426. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 317. iT - 9.23e5T^{2} \)
37 \( 1 - 1.69e3iT - 1.87e6T^{2} \)
41 \( 1 + (-1.74e3 + 1.00e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-938. - 1.62e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (665. - 1.15e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.04e3 + 1.17e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-2.57e3 + 1.48e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.09e3 - 3.63e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-7.66e3 - 4.42e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (6.43e3 - 3.71e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (4.26e3 + 7.37e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-3.80e3 + 2.19e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.01e4T + 4.74e7T^{2} \)
89 \( 1 + (3.23e3 + 1.86e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-845. + 487. i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−17.83379306671571383697027766574, −16.24312906451703060651092732428, −14.60787303984200502110804256909, −13.42032279550802814107230515243, −12.73842414177145403344232658230, −11.12352568571527838586056676856, −8.686077157419286685006895491616, −7.80167613362777740073198103005, −4.82955360170490391756841062855, −2.72191012136388483351886521091, 3.49749324714296198308031145215, 5.43849830909099737289700156126, 7.69003474028073789197735169257, 9.433260812266362297486602270819, 10.84195034270897078838432729082, 13.04817074040827514070763306529, 14.22809552971273279146031053272, 15.10671036601427590808798215689, 15.89590052831446954150401683739, 18.10459057174320015290546572603

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