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Properties

Label 2-19-19.7-c3-0-1
Degree $2$
Conductor $19$
Sign $0.511 - 0.859i$
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  + (1.88 + 3.26i)2-s + (−0.5 − 0.866i)3-s + (−3.11 + 5.39i)4-s + (−2.61 − 4.52i)5-s + (1.88 − 3.26i)6-s − 7.08·7-s + 6.68·8-s + (13 − 22.5i)9-s + (9.86 − 17.0i)10-s − 29.3·11-s + 6.22·12-s + (−35.9 + 62.2i)13-s + (−13.3 − 23.1i)14-s + (−2.61 + 4.52i)15-s + (37.5 + 64.9i)16-s + (−26.1 − 45.2i)17-s + ⋯
L(s)  = 1  + (0.666 + 1.15i)2-s + (−0.0962 − 0.166i)3-s + (−0.389 + 0.674i)4-s + (−0.233 − 0.404i)5-s + (0.128 − 0.222i)6-s − 0.382·7-s + 0.295·8-s + (0.481 − 0.833i)9-s + (0.311 − 0.540i)10-s − 0.803·11-s + 0.149·12-s + (−0.766 + 1.32i)13-s + (−0.255 − 0.442i)14-s + (−0.0449 + 0.0779i)15-s + (0.586 + 1.01i)16-s + (−0.372 − 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.511 - 0.859i$
Analytic conductor: \(1.12103\)
Root analytic conductor: \(1.05879\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :3/2),\ 0.511 - 0.859i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.16637 + 0.663457i\)
\(L(\frac12)\) \(\approx\) \(1.16637 + 0.663457i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-54.8 - 62.0i)T \)
good2 \( 1 + (-1.88 - 3.26i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (2.61 + 4.52i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 7.08T + 343T^{2} \)
11 \( 1 + 29.3T + 1.33e3T^{2} \)
13 \( 1 + (35.9 - 62.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (26.1 + 45.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (36.5 - 63.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-108. + 188. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 14.6T + 2.97e4T^{2} \)
37 \( 1 - 377.T + 5.06e4T^{2} \)
41 \( 1 + (35.2 + 61.0i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-32.8 - 56.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (186. - 322. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-146. + 253. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (268. + 465. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (147. - 255. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (485. - 840. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (397. + 687. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (129. + 224. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-148. - 258. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.41e3T + 5.71e5T^{2} \)
89 \( 1 + (-18.4 + 32.0i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (504. + 873. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−17.97089447571658818504831304294, −16.48012809315948941906589463238, −15.77970378188216994283302599923, −14.48632940699755574378479382272, −13.24555098785439847644393943663, −11.94719232392901472865016607469, −9.673628074594725531367843439653, −7.65917478295079727589729720529, −6.33350131973270507163030434249, −4.52980430317438357480991925445, 2.88838182797516627647302683780, 4.93989408625555646310977864210, 7.60484333006746808795575199932, 10.17650002544382095244424616725, 10.95287816099280060594172338651, 12.58165793606914439086535108391, 13.38825612845085032638649998888, 15.07363342270215208967194595949, 16.42249653414971958645677013100, 18.18732976802048198965475202126

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