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Properties

Label 2-19-19.18-c4-0-4
Degree $2$
Conductor $19$
Sign $-0.595 + 0.803i$
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  − 5.50i·2-s − 4.25i·3-s − 14.3·4-s − 6.22·5-s − 23.4·6-s + 8.36·7-s − 9.27i·8-s + 62.9·9-s + 34.2i·10-s + 128.·11-s + 60.8i·12-s + 101. i·13-s − 46.0i·14-s + 26.4i·15-s − 280.·16-s − 27.5·17-s + ⋯
L(s)  = 1  − 1.37i·2-s − 0.472i·3-s − 0.894·4-s − 0.249·5-s − 0.650·6-s + 0.170·7-s − 0.144i·8-s + 0.776·9-s + 0.342i·10-s + 1.05·11-s + 0.422i·12-s + 0.602i·13-s − 0.235i·14-s + 0.117i·15-s − 1.09·16-s − 0.0952·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.595 + 0.803i$
Analytic conductor: \(1.96402\)
Root analytic conductor: \(1.40143\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :2),\ -0.595 + 0.803i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.566762 - 1.12575i\)
\(L(\frac12)\) \(\approx\) \(0.566762 - 1.12575i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (215. - 289. i)T \)
good2 \( 1 + 5.50iT - 16T^{2} \)
3 \( 1 + 4.25iT - 81T^{2} \)
5 \( 1 + 6.22T + 625T^{2} \)
7 \( 1 - 8.36T + 2.40e3T^{2} \)
11 \( 1 - 128.T + 1.46e4T^{2} \)
13 \( 1 - 101. iT - 2.85e4T^{2} \)
17 \( 1 + 27.5T + 8.35e4T^{2} \)
23 \( 1 - 463.T + 2.79e5T^{2} \)
29 \( 1 - 1.13e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.09e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.85e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.52e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.24e3T + 3.41e6T^{2} \)
47 \( 1 + 1.53e3T + 4.87e6T^{2} \)
53 \( 1 + 855. iT - 7.89e6T^{2} \)
59 \( 1 - 2.66e3iT - 1.21e7T^{2} \)
61 \( 1 + 691.T + 1.38e7T^{2} \)
67 \( 1 - 2.99e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.18e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.25e3T + 2.83e7T^{2} \)
79 \( 1 - 3.43e3iT - 3.89e7T^{2} \)
83 \( 1 - 7.19e3T + 4.74e7T^{2} \)
89 \( 1 + 1.06e4iT - 6.27e7T^{2} \)
97 \( 1 - 3.22e3iT - 8.85e7T^{2} \)
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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

−17.76573235580864746854030479446, −16.09103492777842824970313653543, −14.29513857684770429539288352438, −12.80171011224918103454268168571, −11.93602883722589436693747057571, −10.64607847711212504776886181858, −9.134075433092274029135094880537, −6.95844539013092255267451682695, −3.98684889013541380180422930275, −1.58256861612102069882584949014, 4.52661543118466984280265281182, 6.42450242004923506748029341893, 7.889404077562547067488969293782, 9.468481148024772974935029083733, 11.36544530996201354536132130058, 13.32520667665192961893481826871, 14.90481355015220468058284164611, 15.49039243739655895365458016368, 16.74372118655657124795938320184, 17.65953641644730041384599825595

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