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Properties

Label 2-19-19.18-c6-0-7
Degree $2$
Conductor $19$
Sign $-0.646 - 0.763i$
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  − 12.8i·2-s + 21.6i·3-s − 101.·4-s − 216.·5-s + 278.·6-s − 134.·7-s + 480. i·8-s + 260.·9-s + 2.78e3i·10-s − 610.·11-s − 2.19e3i·12-s − 3.17e3i·13-s + 1.72e3i·14-s − 4.69e3i·15-s − 309.·16-s − 4.96e3·17-s + ⋯
L(s)  = 1  − 1.60i·2-s + 0.801i·3-s − 1.58·4-s − 1.73·5-s + 1.28·6-s − 0.391·7-s + 0.938i·8-s + 0.357·9-s + 2.78i·10-s − 0.458·11-s − 1.26i·12-s − 1.44i·13-s + 0.629i·14-s − 1.39i·15-s − 0.0755·16-s − 1.00·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.646 - 0.763i$
Analytic conductor: \(4.37102\)
Root analytic conductor: \(2.09070\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3),\ -0.646 - 0.763i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.118578 + 0.255834i\)
\(L(\frac12)\) \(\approx\) \(0.118578 + 0.255834i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-4.43e3 - 5.23e3i)T \)
good2 \( 1 + 12.8iT - 64T^{2} \)
3 \( 1 - 21.6iT - 729T^{2} \)
5 \( 1 + 216.T + 1.56e4T^{2} \)
7 \( 1 + 134.T + 1.17e5T^{2} \)
11 \( 1 + 610.T + 1.77e6T^{2} \)
13 \( 1 + 3.17e3iT - 4.82e6T^{2} \)
17 \( 1 + 4.96e3T + 2.41e7T^{2} \)
23 \( 1 + 1.09e4T + 1.48e8T^{2} \)
29 \( 1 + 4.36e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.14e3iT - 8.87e8T^{2} \)
37 \( 1 - 3.22e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.44e3iT - 4.75e9T^{2} \)
43 \( 1 - 6.40e4T + 6.32e9T^{2} \)
47 \( 1 + 9.69e4T + 1.07e10T^{2} \)
53 \( 1 + 5.85e4iT - 2.21e10T^{2} \)
59 \( 1 - 7.88e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.24e4T + 5.15e10T^{2} \)
67 \( 1 + 5.57e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.75e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.78e5T + 1.51e11T^{2} \)
79 \( 1 - 4.41e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.86e5T + 3.26e11T^{2} \)
89 \( 1 - 3.72e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.92e5iT - 8.32e11T^{2} \)
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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

−15.98491042092604775085499439751, −15.33094443114720477523799652684, −13.08608393607379757400132612234, −11.99395405908805705512704952074, −10.86272631184494641193884122917, −9.842190076659624761206405135551, −7.994324279576110339122135648880, −4.39395044549187847461462632202, −3.29611306239370129424488288852, −0.18485158753735972823851866807, 4.42177070626662322550645921197, 6.78731464363435454739132857359, 7.46141468750512795208783108068, 8.791050518811783236346406743259, 11.54041956680443066576989974705, 12.97392321271247763624447806039, 14.39336971059319782454175343614, 15.90720737367743059876856467907, 16.06178295156953636888246356186, 17.90455569609069054570711763604

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