Properties

Label 2-19-19.2-c6-0-3
Degree $2$
Conductor $19$
Sign $0.0884 - 0.996i$
Analytic cond. $4.37102$
Root an. cond. $2.09070$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.3 + 2.18i)2-s + (−27.8 + 33.1i)3-s + (88.0 + 32.0i)4-s + (29.3 − 10.6i)5-s + (−416. + 349. i)6-s + (202. + 351. i)7-s + (323. + 186. i)8-s + (−199. − 1.12e3i)9-s + (386. − 68.1i)10-s + (1.18e3 − 2.05e3i)11-s + (−3.51e3 + 2.02e3i)12-s + (1.02e3 + 1.21e3i)13-s + (1.74e3 + 4.78e3i)14-s + (−462. + 1.27e3i)15-s + (−1.00e3 − 841. i)16-s + (1.00e3 − 5.71e3i)17-s + ⋯
L(s)  = 1  + (1.54 + 0.272i)2-s + (−1.03 + 1.22i)3-s + (1.37 + 0.500i)4-s + (0.234 − 0.0855i)5-s + (−1.92 + 1.61i)6-s + (0.590 + 1.02i)7-s + (0.631 + 0.364i)8-s + (−0.273 − 1.54i)9-s + (0.386 − 0.0681i)10-s + (0.893 − 1.54i)11-s + (−2.03 + 1.17i)12-s + (0.464 + 0.554i)13-s + (0.634 + 1.74i)14-s + (−0.137 + 0.376i)15-s + (−0.244 − 0.205i)16-s + (0.205 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0884 - 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0884 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.0884 - 0.996i$
Analytic conductor: \(4.37102\)
Root analytic conductor: \(2.09070\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3),\ 0.0884 - 0.996i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.89592 + 1.73505i\)
\(L(\frac12)\) \(\approx\) \(1.89592 + 1.73505i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-747. - 6.81e3i)T \)
good2 \( 1 + (-12.3 - 2.18i)T + (60.1 + 21.8i)T^{2} \)
3 \( 1 + (27.8 - 33.1i)T + (-126. - 717. i)T^{2} \)
5 \( 1 + (-29.3 + 10.6i)T + (1.19e4 - 1.00e4i)T^{2} \)
7 \( 1 + (-202. - 351. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-1.18e3 + 2.05e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (-1.02e3 - 1.21e3i)T + (-8.38e5 + 4.75e6i)T^{2} \)
17 \( 1 + (-1.00e3 + 5.71e3i)T + (-2.26e7 - 8.25e6i)T^{2} \)
23 \( 1 + (-3.50e3 - 1.27e3i)T + (1.13e8 + 9.51e7i)T^{2} \)
29 \( 1 + (2.65e4 - 4.67e3i)T + (5.58e8 - 2.03e8i)T^{2} \)
31 \( 1 + (-2.10e4 + 1.21e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + 2.78e4iT - 2.56e9T^{2} \)
41 \( 1 + (4.16e4 - 4.96e4i)T + (-8.24e8 - 4.67e9i)T^{2} \)
43 \( 1 + (-7.21e4 + 2.62e4i)T + (4.84e9 - 4.06e9i)T^{2} \)
47 \( 1 + (1.65e4 + 9.38e4i)T + (-1.01e10 + 3.68e9i)T^{2} \)
53 \( 1 + (-2.93e4 + 8.05e4i)T + (-1.69e10 - 1.42e10i)T^{2} \)
59 \( 1 + (-1.30e5 - 2.30e4i)T + (3.96e10 + 1.44e10i)T^{2} \)
61 \( 1 + (2.87e5 + 1.04e5i)T + (3.94e10 + 3.31e10i)T^{2} \)
67 \( 1 + (4.13e5 - 7.29e4i)T + (8.50e10 - 3.09e10i)T^{2} \)
71 \( 1 + (3.20e4 + 8.80e4i)T + (-9.81e10 + 8.23e10i)T^{2} \)
73 \( 1 + (-2.60e5 - 2.18e5i)T + (2.62e10 + 1.49e11i)T^{2} \)
79 \( 1 + (3.50e5 - 4.17e5i)T + (-4.22e10 - 2.39e11i)T^{2} \)
83 \( 1 + (-5.35e4 - 9.26e4i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (-2.21e5 - 2.63e5i)T + (-8.62e10 + 4.89e11i)T^{2} \)
97 \( 1 + (-2.85e5 - 5.04e4i)T + (7.82e11 + 2.84e11i)T^{2} \)
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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

−16.77495604894976292965381857201, −16.01857662153717930603978754282, −14.89772529043653094480829031065, −13.77647219279232278254027755917, −11.81751498097126695706499913525, −11.36267689022403451407382048846, −9.216049847953102514384162445848, −6.03520347117440110380649740912, −5.30280400669371588552068187200, −3.73871386868884562723597804806, 1.58669378056784463338300793705, 4.45367757503535965499964506601, 6.10135449304585383662503007908, 7.28419386657215616053309261881, 10.74302720406766899964560689814, 11.91175712358342599050556829800, 12.84809975431469831196351460140, 13.79813522192343538354131891038, 15.08279923425236605751598540389, 17.23844292459614430787925095632

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