Properties

Label 2-19-19.6-c1-0-0
Degree $2$
Conductor $19$
Sign $0.500 - 0.865i$
Analytic cond. $0.151715$
Root an. cond. $0.389507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.62i)2-s + (0.613 + 0.223i)3-s + (0.766 − 4.34i)4-s + (−0.233 − 1.32i)5-s + (−1.55 + 0.565i)6-s + (−0.766 + 1.32i)7-s + (3.05 + 5.28i)8-s + (−1.97 − 1.65i)9-s + (2.61 + 2.19i)10-s + (0.592 + 1.02i)11-s + (1.43 − 2.49i)12-s + (−2.55 + 0.929i)13-s + (−0.673 − 3.82i)14-s + (0.152 − 0.866i)15-s + (−6.23 − 2.27i)16-s + (2.97 − 2.49i)17-s + ⋯
L(s)  = 1  + (−1.37 + 1.15i)2-s + (0.354 + 0.128i)3-s + (0.383 − 2.17i)4-s + (−0.104 − 0.593i)5-s + (−0.634 + 0.230i)6-s + (−0.289 + 0.501i)7-s + (1.07 + 1.86i)8-s + (−0.657 − 0.551i)9-s + (0.826 + 0.693i)10-s + (0.178 + 0.309i)11-s + (0.415 − 0.719i)12-s + (−0.708 + 0.257i)13-s + (−0.180 − 1.02i)14-s + (0.0394 − 0.223i)15-s + (−1.55 − 0.567i)16-s + (0.720 − 0.604i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(0.151715\)
Root analytic conductor: \(0.389507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 19,\ (\ :1/2),\ 0.500 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315586 + 0.182066i\)
\(L(\frac12)\) \(\approx\) \(0.315586 + 0.182066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + (-0.819 - 4.28i)T \)
good2 \( 1 + (1.93 - 1.62i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-0.613 - 0.223i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.233 + 1.32i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.766 - 1.32i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.592 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.55 - 0.929i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.97 + 2.49i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.879 - 4.98i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.56 + 2.99i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.91 + 3.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.10T + 37T^{2} \)
41 \( 1 + (-9.38 - 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.51 + 8.57i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.439 - 0.368i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.511 + 2.89i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (3.01 - 2.52i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.784 - 4.44i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.97 + 2.49i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.20 + 6.83i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (5.75 + 2.09i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.21 + 3.35i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (6.15 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.27 + 0.829i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.64 + 4.73i)T + (16.8 - 95.5i)T^{2} \)
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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

−18.64769786126024164034748744806, −17.38903320656237636774339173691, −16.49171658055940940778840062233, −15.34010531092120958757460543924, −14.31303028238350861606711702864, −12.01330198531743448583992967123, −9.789451133347758811425143701092, −8.964246542321939847993066642946, −7.60518620624050872929393087320, −5.78031012960520657860516432631, 2.95600261473892036517366115513, 7.39209480653299014951293223245, 8.755142049334680624551706331763, 10.26542560881370856685877618346, 11.20109023371604474172051081790, 12.72471086044817483942803291073, 14.41031108850262875564062938710, 16.50519445136723328310599643142, 17.46286449521392914520972112470, 18.77735995923535259005713061412

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