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Properties

Label 2-819-117.20-c1-0-22
Degree $2$
Conductor $819$
Sign $0.889 + 0.457i$
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  + (−2.46 − 0.660i)2-s + (−1.69 + 0.367i)3-s + (3.91 + 2.26i)4-s + (−1.36 − 0.366i)5-s + (4.41 + 0.210i)6-s + (0.707 − 0.707i)7-s + (−4.55 − 4.55i)8-s + (2.72 − 1.24i)9-s + (3.13 + 1.80i)10-s + (−0.0907 + 0.338i)11-s + (−7.45 − 2.38i)12-s + (1.89 + 3.06i)13-s + (−2.21 + 1.27i)14-s + (2.45 + 0.116i)15-s + (3.69 + 6.40i)16-s + (0.610 + 1.05i)17-s + ⋯
L(s)  = 1  + (−1.74 − 0.467i)2-s + (−0.977 + 0.212i)3-s + (1.95 + 1.13i)4-s + (−0.611 − 0.163i)5-s + (1.80 + 0.0861i)6-s + (0.267 − 0.267i)7-s + (−1.60 − 1.60i)8-s + (0.909 − 0.415i)9-s + (0.990 + 0.571i)10-s + (−0.0273 + 0.102i)11-s + (−2.15 − 0.688i)12-s + (0.526 + 0.850i)13-s + (−0.591 + 0.341i)14-s + (0.632 + 0.0302i)15-s + (0.924 + 1.60i)16-s + (0.147 + 0.256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.889 + 0.457i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (722, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.889 + 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371259 - 0.0898719i\)
\(L(\frac12)\) \(\approx\) \(0.371259 - 0.0898719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 - 0.367i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-1.89 - 3.06i)T \)
good2 \( 1 + (2.46 + 0.660i)T + (1.73 + i)T^{2} \)
5 \( 1 + (1.36 + 0.366i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0907 - 0.338i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.610 - 1.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.59 + 5.95i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.26T + 23T^{2} \)
29 \( 1 + (2.82 - 1.63i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.04 - 7.62i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.237 - 0.886i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.31 + 8.31i)T - 41iT^{2} \)
43 \( 1 - 5.43iT - 43T^{2} \)
47 \( 1 + (0.551 - 0.147i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 7.24iT - 53T^{2} \)
59 \( 1 + (-12.7 + 3.42i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 1.81T + 61T^{2} \)
67 \( 1 + (4.51 + 4.51i)T + 67iT^{2} \)
71 \( 1 + (0.512 + 0.137i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.14 + 5.14i)T - 73iT^{2} \)
79 \( 1 + (-1.90 + 3.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.09 + 11.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-13.5 + 3.62i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.09 + 7.09i)T + 97iT^{2} \)
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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

−10.26587050985831327086398558408, −9.333155760333563413250454090520, −8.730314110147051879736714876721, −7.64144452438971147053220249949, −7.08654423773788312252083887499, −6.12080526086049598108467351524, −4.68898120391228126813539489045, −3.60522439515420466786585868543, −1.89919249195761274866398882984, −0.66271113794962543050632634818, 0.70779468120468036561056349315, 2.01647331653021766364418147299, 3.89217436500984185386649138935, 5.63692174248640784702120606166, 5.99354186904065045292253262089, 7.19512598270214758898924886375, 7.84733865234728311096745830469, 8.296482035320953389836014411167, 9.654693665235048707508533251782, 10.08683128424846597471595832256

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