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Properties

Label 2-19-19.11-c3-0-1
Degree $2$
Conductor $19$
Sign $0.205 - 0.978i$
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  + (−0.5 + 0.866i)2-s + (−3.70 + 6.42i)3-s + (3.5 + 6.06i)4-s + (7.20 − 12.4i)5-s + (−3.70 − 6.42i)6-s + 0.416·7-s − 15·8-s + (−14 − 24.2i)9-s + (7.20 + 12.4i)10-s + 65.9·11-s − 51.9·12-s + (−22.6 − 39.2i)13-s + (−0.208 + 0.360i)14-s + (53.4 + 92.5i)15-s + (−20.5 + 35.5i)16-s + (−1.66 + 2.88i)17-s + ⋯
L(s)  = 1  + (−0.176 + 0.306i)2-s + (−0.713 + 1.23i)3-s + (0.437 + 0.757i)4-s + (0.644 − 1.11i)5-s + (−0.252 − 0.437i)6-s + 0.0224·7-s − 0.662·8-s + (−0.518 − 0.898i)9-s + (0.227 + 0.394i)10-s + 1.80·11-s − 1.24·12-s + (−0.483 − 0.837i)13-s + (−0.00397 + 0.00688i)14-s + (0.920 + 1.59i)15-s + (−0.320 + 0.554i)16-s + (−0.0237 + 0.0411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.735827 + 0.597412i\)
\(L(\frac12)\) \(\approx\) \(0.735827 + 0.597412i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (82.3 - 8.58i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (3.70 - 6.42i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-7.20 + 12.4i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 0.416T + 343T^{2} \)
11 \( 1 - 65.9T + 1.33e3T^{2} \)
13 \( 1 + (22.6 + 39.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (1.66 - 2.88i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (54.4 + 94.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (29.5 + 51.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 184.T + 2.97e4T^{2} \)
37 \( 1 - 142.T + 5.06e4T^{2} \)
41 \( 1 + (-87.5 + 151. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (222. - 385. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (87.1 + 150. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-117. - 202. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (76.2 - 131. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (28.9 + 50.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-418. - 725. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (263. - 456. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (109. - 190. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-245. + 425. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 692.T + 5.71e5T^{2} \)
89 \( 1 + (413. + 716. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-366. + 634. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−17.40923494721151964661117032802, −17.00245295547200147172759254085, −16.18970928024624579260373007548, −14.85024118861606704928053979165, −12.73467316312421954407933360268, −11.57149970507948445421497158817, −9.863506073124870342877110887101, −8.621413490449885660475474669216, −6.15725356574837595106834497624, −4.35393353901368161250089349715, 1.80707349095875850976934547520, 6.28325877538933314029638690587, 6.79355637724893168100621851114, 9.639107111983385708910907159760, 11.18219070282663027290289746649, 12.02385442510357635313138596983, 13.90417977939677390310978154853, 14.82549206667388121647402799032, 16.98634599910734003893039707091, 17.97252127723841563049350418388

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