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Properties

Label 2-19-19.18-c6-0-6
Degree $2$
Conductor $19$
Sign $-0.788 + 0.615i$
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  − 5.43i·2-s − 33.7i·3-s + 34.4·4-s − 51.7·5-s − 183.·6-s − 313.·7-s − 535. i·8-s − 409.·9-s + 281. i·10-s + 460.·11-s − 1.16e3i·12-s − 177. i·13-s + 1.70e3i·14-s + 1.74e3i·15-s − 702.·16-s + 6.12e3·17-s + ⋯
L(s)  = 1  − 0.679i·2-s − 1.24i·3-s + 0.538·4-s − 0.414·5-s − 0.848·6-s − 0.913·7-s − 1.04i·8-s − 0.561·9-s + 0.281i·10-s + 0.346·11-s − 0.673i·12-s − 0.0806i·13-s + 0.620i·14-s + 0.517i·15-s − 0.171·16-s + 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.788 + 0.615i$
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: \((2,\ 19,\ (\ :3),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.484838 - 1.40917i\)
\(L(\frac12)\) \(\approx\) \(0.484838 - 1.40917i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-5.40e3 + 4.22e3i)T \)
good2 \( 1 + 5.43iT - 64T^{2} \)
3 \( 1 + 33.7iT - 729T^{2} \)
5 \( 1 + 51.7T + 1.56e4T^{2} \)
7 \( 1 + 313.T + 1.17e5T^{2} \)
11 \( 1 - 460.T + 1.77e6T^{2} \)
13 \( 1 + 177. iT - 4.82e6T^{2} \)
17 \( 1 - 6.12e3T + 2.41e7T^{2} \)
23 \( 1 - 1.09e4T + 1.48e8T^{2} \)
29 \( 1 + 1.11e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.59e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.65e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.25e5iT - 4.75e9T^{2} \)
43 \( 1 + 2.06e4T + 6.32e9T^{2} \)
47 \( 1 - 1.50e5T + 1.07e10T^{2} \)
53 \( 1 + 1.13e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.31e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.14e5T + 5.15e10T^{2} \)
67 \( 1 + 3.79e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.18e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.79e5T + 1.51e11T^{2} \)
79 \( 1 - 3.31e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.91e5T + 3.26e11T^{2} \)
89 \( 1 - 2.68e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.70e6iT - 8.32e11T^{2} \)
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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

−16.73052532919690473442725124543, −15.43384458449031058156060737043, −13.50957618631671262035912503376, −12.44008159439054312373548231632, −11.59816454187481519475549671690, −9.832875960469118930546383095601, −7.59504244092420670711554032230, −6.45351999736207093452659194072, −3.09042890455584303974679210582, −1.08239387524706746849662190502, 3.54878702769351381433696372813, 5.62576795645179455268057380603, 7.41328193950356372890368011949, 9.332125904974229929229543252023, 10.65457269634270646840437367687, 12.14764733692745372699058475318, 14.31655486454879408817838490893, 15.47725732244245194074905857569, 16.16635986433140408010315236776, 16.98793190436024433574820184543

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