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Properties

Label 2-19-19.18-c6-0-5
Degree $2$
Conductor $19$
Sign $0.881 - 0.472i$
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  + 8.08i·2-s − 27.2i·3-s − 1.29·4-s + 162.·5-s + 219.·6-s − 68.0·7-s + 506. i·8-s − 11.4·9-s + 1.31e3i·10-s + 838.·11-s + 35.2i·12-s − 2.03e3i·13-s − 550. i·14-s − 4.43e3i·15-s − 4.17e3·16-s + 1.26e3·17-s + ⋯
L(s)  = 1  + 1.01i·2-s − 1.00i·3-s − 0.0202·4-s + 1.30·5-s + 1.01·6-s − 0.198·7-s + 0.989i·8-s − 0.0157·9-s + 1.31i·10-s + 0.629·11-s + 0.0204i·12-s − 0.925i·13-s − 0.200i·14-s − 1.31i·15-s − 1.01·16-s + 0.258·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.881 - 0.472i$
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: \((2,\ 19,\ (\ :3),\ 0.881 - 0.472i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.85796 + 0.466147i\)
\(L(\frac12)\) \(\approx\) \(1.85796 + 0.466147i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (6.04e3 - 3.23e3i)T \)
good2 \( 1 - 8.08iT - 64T^{2} \)
3 \( 1 + 27.2iT - 729T^{2} \)
5 \( 1 - 162.T + 1.56e4T^{2} \)
7 \( 1 + 68.0T + 1.17e5T^{2} \)
11 \( 1 - 838.T + 1.77e6T^{2} \)
13 \( 1 + 2.03e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.26e3T + 2.41e7T^{2} \)
23 \( 1 + 2.23e4T + 1.48e8T^{2} \)
29 \( 1 + 2.15e3iT - 5.94e8T^{2} \)
31 \( 1 - 4.25e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.60e4iT - 2.56e9T^{2} \)
41 \( 1 + 7.83e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.94e4T + 6.32e9T^{2} \)
47 \( 1 + 3.75e4T + 1.07e10T^{2} \)
53 \( 1 + 1.66e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.81e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.73e5T + 5.15e10T^{2} \)
67 \( 1 - 2.73e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.38e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.48e5T + 1.51e11T^{2} \)
79 \( 1 + 5.74e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.44e5T + 3.26e11T^{2} \)
89 \( 1 - 1.32e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.42e5iT - 8.32e11T^{2} \)
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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

−17.43023552606105111452301423444, −16.11869778388891111402485931396, −14.54865364134865135651114384297, −13.55915446022866599906384711563, −12.25945621300088048418208325546, −10.15658360772215550073011522256, −8.229674150531604528419300623881, −6.72703384211441407156213490463, −5.80111195165395356311207723466, −1.89969318899710707443652956283, 1.98851904663012117812876644586, 4.09188953663521930284471797219, 6.33798897140307399827236074430, 9.441884061930544947808277625209, 10.01888079072316349307481374900, 11.35709867334715433612465211013, 12.91700647428766662756691542256, 14.32792078656118054270439593185, 15.90311192257975460864185063930, 17.00614615262526667560346608544

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