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Properties

Label 2-19-19.8-c4-0-0
Degree $2$
Conductor $19$
Sign $-0.474 - 0.880i$
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  + (−5.19 − 3.00i)2-s + (−1.61 − 0.934i)3-s + (10.0 + 17.3i)4-s + (−20.3 + 35.1i)5-s + (5.61 + 9.71i)6-s − 11.8·7-s − 24.0i·8-s + (−38.7 − 67.1i)9-s + (211. − 121. i)10-s − 91.6·11-s − 37.4i·12-s + (−81.2 + 46.8i)13-s + (61.6 + 35.5i)14-s + (65.7 − 37.9i)15-s + (87.8 − 152. i)16-s + (−205. + 356. i)17-s + ⋯
L(s)  = 1  + (−1.29 − 0.750i)2-s + (−0.179 − 0.103i)3-s + (0.625 + 1.08i)4-s + (−0.812 + 1.40i)5-s + (0.155 + 0.269i)6-s − 0.242·7-s − 0.376i·8-s + (−0.478 − 0.828i)9-s + (2.11 − 1.21i)10-s − 0.757·11-s − 0.259i·12-s + (−0.480 + 0.277i)13-s + (0.314 + 0.181i)14-s + (0.292 − 0.168i)15-s + (0.343 − 0.594i)16-s + (−0.711 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.474 - 0.880i$
Analytic conductor: \(1.96402\)
Root analytic conductor: \(1.40143\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :2),\ -0.474 - 0.880i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0830246 + 0.139003i\)
\(L(\frac12)\) \(\approx\) \(0.0830246 + 0.139003i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-333. + 139. i)T \)
good2 \( 1 + (5.19 + 3.00i)T + (8 + 13.8i)T^{2} \)
3 \( 1 + (1.61 + 0.934i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (20.3 - 35.1i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + 11.8T + 2.40e3T^{2} \)
11 \( 1 + 91.6T + 1.46e4T^{2} \)
13 \( 1 + (81.2 - 46.8i)T + (1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (205. - 356. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
23 \( 1 + (168. + 291. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-932. + 538. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 - 1.78e3iT - 9.23e5T^{2} \)
37 \( 1 + 240. iT - 1.87e6T^{2} \)
41 \( 1 + (1.87e3 + 1.08e3i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.14e3 - 1.97e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-818. - 1.41e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-108. + 62.8i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-3.42e3 - 1.97e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.09e3 + 1.89e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.58e3 + 915. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-994. - 574. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (2.47e3 - 4.28e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-103. - 60.0i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 1.03e4T + 4.74e7T^{2} \)
89 \( 1 + (661. - 381. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-3.75e3 - 2.16e3i)T + (4.42e7 + 7.66e7i)T^{2} \)
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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

−18.24622682654572471878384848577, −17.53208787730263518250061296214, −15.76303639405729301277181898961, −14.46582893417409744237921043078, −12.14875737902339890519225755935, −11.09542762869046920251057648958, −10.09497264695077601629590084643, −8.362855958937209497007742058166, −6.83304536315025630997585177445, −2.98389855025897535345658538559, 0.21847194291957140268237958099, 5.15063494567016896416000281776, 7.56626140776977057432150252377, 8.512942883739957345312812283002, 9.876986267810012400857926595958, 11.69217616119373879112589849814, 13.30991044939972886896053486393, 15.60468000777892511168797297223, 16.22785535543157572846909267550, 17.07274630324654673341624229846

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