Properties

Label 2-19-19.9-c3-0-3
Degree $2$
Conductor $19$
Sign $-0.476 + 0.879i$
Analytic cond. $1.12103$
Root an. cond. $1.05879$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.851 − 4.82i)2-s + (0.458 − 0.384i)3-s + (−15.0 + 5.49i)4-s + (6.94 + 2.52i)5-s + (−2.24 − 1.88i)6-s + (14.5 − 25.2i)7-s + (19.7 + 34.2i)8-s + (−4.62 + 26.2i)9-s + (6.29 − 35.7i)10-s + (26.3 + 45.7i)11-s + (−4.80 + 8.32i)12-s + (−30.9 − 26.0i)13-s + (−134. − 48.8i)14-s + (4.15 − 1.51i)15-s + (50.0 − 41.9i)16-s + (−0.0277 − 0.157i)17-s + ⋯
L(s)  = 1  + (−0.301 − 1.70i)2-s + (0.0882 − 0.0740i)3-s + (−1.88 + 0.686i)4-s + (0.621 + 0.226i)5-s + (−0.153 − 0.128i)6-s + (0.786 − 1.36i)7-s + (0.872 + 1.51i)8-s + (−0.171 + 0.971i)9-s + (0.199 − 1.12i)10-s + (0.723 + 1.25i)11-s + (−0.115 + 0.200i)12-s + (−0.661 − 0.554i)13-s + (−2.56 − 0.933i)14-s + (0.0715 − 0.0260i)15-s + (0.781 − 0.655i)16-s + (−0.000395 − 0.00224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ -0.476 + 0.879i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.497272 - 0.834913i\)
\(L(\frac12)\) \(\approx\) \(0.497272 - 0.834913i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (13.4 - 81.7i)T \)
good2 \( 1 + (0.851 + 4.82i)T + (-7.51 + 2.73i)T^{2} \)
3 \( 1 + (-0.458 + 0.384i)T + (4.68 - 26.5i)T^{2} \)
5 \( 1 + (-6.94 - 2.52i)T + (95.7 + 80.3i)T^{2} \)
7 \( 1 + (-14.5 + 25.2i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-26.3 - 45.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (30.9 + 26.0i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (0.0277 + 0.157i)T + (-4.61e3 + 1.68e3i)T^{2} \)
23 \( 1 + (39.4 - 14.3i)T + (9.32e3 - 7.82e3i)T^{2} \)
29 \( 1 + (-11.8 + 66.9i)T + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-108. + 187. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 288.T + 5.06e4T^{2} \)
41 \( 1 + (24.4 - 20.4i)T + (1.19e4 - 6.78e4i)T^{2} \)
43 \( 1 + (-128. - 46.8i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (32.7 - 185. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + (31.5 - 11.4i)T + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (-50.5 - 286. i)T + (-1.92e5 + 7.02e4i)T^{2} \)
61 \( 1 + (10.9 - 4.00i)T + (1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (-61.8 + 350. i)T + (-2.82e5 - 1.02e5i)T^{2} \)
71 \( 1 + (237. + 86.4i)T + (2.74e5 + 2.30e5i)T^{2} \)
73 \( 1 + (619. - 519. i)T + (6.75e4 - 3.83e5i)T^{2} \)
79 \( 1 + (-540. + 453. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-608. + 1.05e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-895. - 751. i)T + (1.22e5 + 6.94e5i)T^{2} \)
97 \( 1 + (69.0 + 391. i)T + (-8.57e5 + 3.12e5i)T^{2} \)
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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

−17.70934285464968449903553812897, −17.15742414484962570428447036630, −14.37519980642387978300290591800, −13.43027296057373223494949327059, −11.99386035691662806676110181306, −10.56446462903933111489179809580, −9.884690990208406748667191346597, −7.78906971110019606873340229109, −4.38544348485857710567153099270, −1.89422877514160191127270379411, 5.30795312797621962920238450076, 6.51451448215949238147278493836, 8.622860819271479278195669629169, 9.199162985915140776562811729272, 11.87916840916491996052765983677, 13.93016927599919676452666539247, 14.78752822705444048120796019113, 15.84607431856518601505961073891, 17.21784062313970370627199098757, 17.90368293811860468012822633383

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