Properties

Label 2-19-19.7-c3-0-3
Degree $2$
Conductor $19$
Sign $-0.805 + 0.592i$
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  + (−2.38 − 4.13i)2-s + (−0.5 − 0.866i)3-s + (−7.38 + 12.7i)4-s + (−6.88 − 11.9i)5-s + (−2.38 + 4.13i)6-s + 27.0·7-s + 32.3·8-s + (13 − 22.5i)9-s + (−32.8 + 56.9i)10-s − 3.68·11-s + 14.7·12-s + (−14.5 + 25.2i)13-s + (−64.6 − 111. i)14-s + (−6.88 + 11.9i)15-s + (−18.0 − 31.2i)16-s + (63.6 + 110. i)17-s + ⋯
L(s)  = 1  + (−0.843 − 1.46i)2-s + (−0.0962 − 0.166i)3-s + (−0.923 + 1.59i)4-s + (−0.615 − 1.06i)5-s + (−0.162 + 0.281i)6-s + 1.46·7-s + 1.42·8-s + (0.481 − 0.833i)9-s + (−1.03 + 1.79i)10-s − 0.100·11-s + 0.355·12-s + (−0.310 + 0.538i)13-s + (−1.23 − 2.13i)14-s + (−0.118 + 0.205i)15-s + (−0.281 − 0.487i)16-s + (0.907 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $-0.805 + 0.592i$
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: \((2,\ 19,\ (\ :3/2),\ -0.805 + 0.592i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.212848 - 0.648439i\)
\(L(\frac12)\) \(\approx\) \(0.212848 - 0.648439i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (26.3 + 78.5i)T \)
good2 \( 1 + (2.38 + 4.13i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (6.88 + 11.9i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 27.0T + 343T^{2} \)
11 \( 1 + 3.68T + 1.33e3T^{2} \)
13 \( 1 + (14.5 - 25.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-63.6 - 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-36.0 + 62.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (66.3 - 114. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 36.6T + 2.97e4T^{2} \)
37 \( 1 - 70.2T + 5.06e4T^{2} \)
41 \( 1 + (26.7 + 46.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-122. - 212. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (19.4 - 33.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (276. - 479. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-166. - 289. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (117. - 204. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-207. + 358. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (384. + 665. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-246. - 426. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-16.5 - 28.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 41.7T + 5.71e5T^{2} \)
89 \( 1 + (318. - 552. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-666. - 1.15e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−17.78519776323034299482155693929, −16.88362379104491525374312874736, −14.91646054748791255515594991777, −12.73219504325400140510692688207, −11.99535748711843374482209828702, −10.81353709486644099513906672459, −9.090306010611435308001886336950, −8.040001423454698361626527504655, −4.33648410328182797990327425140, −1.28359854954818164315969320333, 5.21346500966785888043123614810, 7.37076504360630409060805599513, 7.966303031422739308937698687555, 10.04640093347099595472234895511, 11.39059954791368989518783844836, 14.14320385298325382452985799675, 14.94980577418230059077860698572, 15.98161036395041775163673283248, 17.23688265764974327129179194567, 18.37081240776476443758851422985

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