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Properties

Label 2-819-91.31-c1-0-20
Degree $2$
Conductor $819$
Sign $-0.170 + 0.985i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (−1.53 − 0.411i)2-s + (0.454 + 0.262i)4-s + (0.0206 − 0.0769i)5-s + (−2.16 + 1.52i)7-s + (1.65 + 1.65i)8-s + (−0.0632 + 0.109i)10-s + (−4.08 + 1.09i)11-s + (−0.565 + 3.56i)13-s + (3.94 − 1.44i)14-s + (−2.38 − 4.13i)16-s + (2.90 − 5.02i)17-s + (1.36 − 5.11i)19-s + (0.0295 − 0.0295i)20-s + 6.71·22-s + (0.755 − 0.436i)23-s + ⋯
L(s)  = 1  + (−1.08 − 0.290i)2-s + (0.227 + 0.131i)4-s + (0.00921 − 0.0343i)5-s + (−0.818 + 0.575i)7-s + (0.585 + 0.585i)8-s + (−0.0200 + 0.0346i)10-s + (−1.23 + 0.329i)11-s + (−0.156 + 0.987i)13-s + (1.05 − 0.386i)14-s + (−0.596 − 1.03i)16-s + (0.704 − 1.21i)17-s + (0.314 − 1.17i)19-s + (0.00661 − 0.00661i)20-s + 1.43·22-s + (0.157 − 0.0909i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.170 + 0.985i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.170 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280094 - 0.332692i\)
\(L(\frac12)\) \(\approx\) \(0.280094 - 0.332692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.16 - 1.52i)T \)
13 \( 1 + (0.565 - 3.56i)T \)
good2 \( 1 + (1.53 + 0.411i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.0206 + 0.0769i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.08 - 1.09i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.90 + 5.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.36 + 5.11i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.755 + 0.436i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.362T + 29T^{2} \)
31 \( 1 + (1.34 - 0.361i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.00 + 3.76i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.70 + 7.70i)T + 41iT^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 + (-2.79 - 0.748i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.26 + 9.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.573 - 2.14i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.63 + 2.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.61 + 9.76i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.65 - 3.65i)T - 71iT^{2} \)
73 \( 1 + (3.08 + 11.4i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.27 + 7.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.91 - 4.91i)T + 83iT^{2} \)
89 \( 1 + (-7.78 - 2.08i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−9.812254159737534607752442708775, −9.230611628143356659066389196285, −8.652137750412831046913346445123, −7.45422458907522370850708395952, −6.93603249901804397770539409412, −5.42412221467340503396067646341, −4.81119331667775108206220872713, −3.07116836272369116184954751076, −2.13829052733603810154929323885, −0.37134742496044003964777254698, 1.08146910189746664730126813038, 2.97845291701405458051289059334, 3.94929634597170319738874177462, 5.36132972426907067882861735976, 6.30304313876420938817701213272, 7.37066967436059352519829667493, 8.005907462869784614357827719075, 8.589204006759280167545462783304, 9.802162618303872953584285511666, 10.34663972628446483823324708149

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