Properties

Label 2-19-19.4-c3-0-1
Degree $2$
Conductor $19$
Sign $0.716 - 0.697i$
Analytic cond. $1.12103$
Root an. cond. $1.05879$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.579 + 0.210i)2-s + (1.63 + 9.27i)3-s + (−5.83 − 4.89i)4-s + (9.88 − 8.29i)5-s + (−1.00 + 5.71i)6-s + (6.05 − 10.4i)7-s + (−4.81 − 8.33i)8-s + (−57.9 + 21.0i)9-s + (7.46 − 2.71i)10-s + (1.51 + 2.61i)11-s + (35.8 − 62.1i)12-s + (−2.02 + 11.4i)13-s + (5.71 − 4.79i)14-s + (93.0 + 78.0i)15-s + (9.55 + 54.1i)16-s + (−104. − 38.0i)17-s + ⋯
L(s)  = 1  + (0.204 + 0.0745i)2-s + (0.314 + 1.78i)3-s + (−0.729 − 0.612i)4-s + (0.883 − 0.741i)5-s + (−0.0685 + 0.388i)6-s + (0.327 − 0.566i)7-s + (−0.212 − 0.368i)8-s + (−2.14 + 0.781i)9-s + (0.236 − 0.0859i)10-s + (0.0414 + 0.0717i)11-s + (0.863 − 1.49i)12-s + (−0.0432 + 0.245i)13-s + (0.109 − 0.0916i)14-s + (1.60 + 1.34i)15-s + (0.149 + 0.846i)16-s + (−1.49 − 0.542i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ 0.716 - 0.697i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11107 + 0.451613i\)
\(L(\frac12)\) \(\approx\) \(1.11107 + 0.451613i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-82.6 - 5.09i)T \)
good2 \( 1 + (-0.579 - 0.210i)T + (6.12 + 5.14i)T^{2} \)
3 \( 1 + (-1.63 - 9.27i)T + (-25.3 + 9.23i)T^{2} \)
5 \( 1 + (-9.88 + 8.29i)T + (21.7 - 123. i)T^{2} \)
7 \( 1 + (-6.05 + 10.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-1.51 - 2.61i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (2.02 - 11.4i)T + (-2.06e3 - 751. i)T^{2} \)
17 \( 1 + (104. + 38.0i)T + (3.76e3 + 3.15e3i)T^{2} \)
23 \( 1 + (0.581 + 0.487i)T + (2.11e3 + 1.19e4i)T^{2} \)
29 \( 1 + (96.6 - 35.1i)T + (1.86e4 - 1.56e4i)T^{2} \)
31 \( 1 + (59.9 - 103. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 336.T + 5.06e4T^{2} \)
41 \( 1 + (17.2 + 97.7i)T + (-6.47e4 + 2.35e4i)T^{2} \)
43 \( 1 + (-88.1 + 73.9i)T + (1.38e4 - 7.82e4i)T^{2} \)
47 \( 1 + (-298. + 108. i)T + (7.95e4 - 6.67e4i)T^{2} \)
53 \( 1 + (244. + 205. i)T + (2.58e4 + 1.46e5i)T^{2} \)
59 \( 1 + (-225. - 82.0i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (166. + 139. i)T + (3.94e4 + 2.23e5i)T^{2} \)
67 \( 1 + (510. - 185. i)T + (2.30e5 - 1.93e5i)T^{2} \)
71 \( 1 + (-166. + 139. i)T + (6.21e4 - 3.52e5i)T^{2} \)
73 \( 1 + (99.2 + 562. i)T + (-3.65e5 + 1.33e5i)T^{2} \)
79 \( 1 + (-185. - 1.05e3i)T + (-4.63e5 + 1.68e5i)T^{2} \)
83 \( 1 + (315. - 546. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (23.4 - 132. i)T + (-6.62e5 - 2.41e5i)T^{2} \)
97 \( 1 + (-859. - 312. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−17.87738343950880896643194610084, −16.74660068488962049713388767220, −15.55741824976371235771066126492, −14.31437017640345369088031410350, −13.46834804406925733367034118405, −10.90114475097715645673970018449, −9.627613425538338900979465487093, −8.990005529828433178660473681652, −5.37910527992928316945820457893, −4.32819504465645121356367857857, 2.48099309715001762746499871625, 6.05004923023977833302968219967, 7.64933940408179660916685891502, 9.031602240867070571421724529449, 11.55726474177970449670943983123, 12.91848157435819790628321412440, 13.62886766726311204760573949067, 14.66982722899333181059117633242, 17.36255154994526994279959887362, 18.06008442891240130188924874527

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