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Properties

Label 2-19-19.18-c2-0-0
Degree $2$
Conductor $19$
Sign $0.315 - 0.948i$
Analytic cond. $0.517712$
Root an. cond. $0.719522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.60i·2-s − 3.60i·3-s − 8.99·4-s + 4·5-s + 12.9·6-s − 5·7-s − 18.0i·8-s − 3.99·9-s + 14.4i·10-s − 10·11-s + 32.4i·12-s + 3.60i·13-s − 18.0i·14-s − 14.4i·15-s + 28.9·16-s + 15·17-s + ⋯
L(s)  = 1  + 1.80i·2-s − 1.20i·3-s − 2.24·4-s + 0.800·5-s + 2.16·6-s − 0.714·7-s − 2.25i·8-s − 0.444·9-s + 1.44i·10-s − 0.909·11-s + 2.70i·12-s + 0.277i·13-s − 1.28i·14-s − 0.961i·15-s + 1.81·16-s + 0.882·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(0.517712\)
Root analytic conductor: \(0.719522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :1),\ 0.315 - 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.654546 + 0.472000i\)
\(L(\frac12)\) \(\approx\) \(0.654546 + 0.472000i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (6 - 18.0i)T \)
good2 \( 1 - 3.60iT - 4T^{2} \)
3 \( 1 + 3.60iT - 9T^{2} \)
5 \( 1 - 4T + 25T^{2} \)
7 \( 1 + 5T + 49T^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 - 3.60iT - 169T^{2} \)
17 \( 1 - 15T + 289T^{2} \)
23 \( 1 - 35T + 529T^{2} \)
29 \( 1 - 18.0iT - 841T^{2} \)
31 \( 1 + 36.0iT - 961T^{2} \)
37 \( 1 + 21.6iT - 1.36e3T^{2} \)
41 \( 1 - 36.0iT - 1.68e3T^{2} \)
43 \( 1 + 20T + 1.84e3T^{2} \)
47 \( 1 - 10T + 2.20e3T^{2} \)
53 \( 1 + 75.7iT - 2.80e3T^{2} \)
59 \( 1 - 18.0iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 - 39.6iT - 4.48e3T^{2} \)
71 \( 1 - 108. iT - 5.04e3T^{2} \)
73 \( 1 - 105T + 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 + 40T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 122. iT - 9.40e3T^{2} \)
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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

−18.32800620868143821671096034871, −17.21035229029363237188423869982, −16.21063509210890520760409018508, −14.71293824049643278009270419978, −13.47082475202125214301786291244, −12.80929082667917783009386997073, −9.709875664515623190473484465539, −8.008777466551688594985583029709, −6.80851237079612159424138322129, −5.65486461226492922850288157593, 3.10819521664665256600463056824, 5.02070792807218811212169335008, 9.163442841217987117557950293699, 10.05941444649321422619702072719, 10.84934664941281548929337477819, 12.64065192308747340816046109002, 13.66465521900764779981549730018, 15.44395203851106089460354228870, 17.08583885424158940884764644480, 18.45697148127112063319969776997

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