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Properties

Label 2-19-19.11-c3-0-2
Degree $2$
Conductor $19$
Sign $0.992 + 0.120i$
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 + (−0.208 + 0.360i)5-s + (3.70 + 6.42i)6-s − 14.4·7-s − 15·8-s + (−14 − 24.2i)9-s + (−0.208 − 0.360i)10-s − 37.9·11-s + 51.9·12-s + (36.6 + 63.5i)13-s + (7.20 − 12.4i)14-s + (1.54 + 2.67i)15-s + (−20.5 + 35.5i)16-s + (57.6 − 99.8i)17-s + ⋯
L(s)  = 1  + (−0.176 + 0.306i)2-s + (0.713 − 1.23i)3-s + (0.437 + 0.757i)4-s + (−0.0186 + 0.0322i)5-s + (0.252 + 0.437i)6-s − 0.778·7-s − 0.662·8-s + (−0.518 − 0.898i)9-s + (−0.00658 − 0.0113i)10-s − 1.03·11-s + 1.24·12-s + (0.782 + 1.35i)13-s + (0.137 − 0.238i)14-s + (0.0265 + 0.0460i)15-s + (−0.320 + 0.554i)16-s + (0.822 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.992 + 0.120i$
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: \((2,\ 19,\ (\ :3/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.16326 - 0.0704157i\)
\(L(\frac12)\) \(\approx\) \(1.16326 - 0.0704157i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (15.6 + 81.3i)T \)
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 \( 1 + (0.208 - 0.360i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + 14.4T + 343T^{2} \)
11 \( 1 + 37.9T + 1.33e3T^{2} \)
13 \( 1 + (-36.6 - 63.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-57.6 + 99.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (2.54 + 4.40i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (81.4 + 141. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 81.0T + 2.97e4T^{2} \)
37 \( 1 - 39.0T + 5.06e4T^{2} \)
41 \( 1 + (164. - 285. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-88.7 + 153. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-24.1 - 41.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-324. - 562. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-20.2 + 34.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (244. + 422. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (48.6 + 84.1i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-47.7 + 82.6i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (65.2 - 113. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (169. - 293. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + (-357. - 619. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (93.4 - 161. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−18.31746274463354472166707796772, −16.70477352897153017797265422343, −15.61956858362561245054527816768, −13.74907908827573654150726878296, −12.92528042527716021954554939053, −11.59180350846452901703352689354, −9.095862318629453283956321728588, −7.66963732698634338869232858888, −6.67815208648970580659865166938, −2.81080177139999889406601552746, 3.28410990329869220790991015590, 5.73887525915786512917881324531, 8.437267932235750411788221689331, 10.17266619050044811358202967564, 10.45447838680352329623674941604, 12.76643311611987639263565177189, 14.58065532351422642765923631671, 15.47735260698737009456932652187, 16.29719430542360245890068302798, 18.38623280751933292867241115970

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