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Properties

Label 2-819-91.45-c1-0-21
Degree $2$
Conductor $819$
Sign $0.818 - 0.574i$
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  + (0.411 − 0.411i)2-s + 1.66i·4-s + (0.180 − 0.672i)5-s + (2.60 + 0.483i)7-s + (1.50 + 1.50i)8-s + (−0.202 − 0.351i)10-s + (0.230 − 0.859i)11-s + (0.659 + 3.54i)13-s + (1.27 − 0.871i)14-s − 2.08·16-s + 0.0460·17-s + (−0.843 + 0.225i)19-s + (1.11 + 0.299i)20-s + (−0.258 − 0.448i)22-s − 3.19i·23-s + ⋯
L(s)  = 1  + (0.291 − 0.291i)2-s + 0.830i·4-s + (0.0806 − 0.300i)5-s + (0.983 + 0.182i)7-s + (0.532 + 0.532i)8-s + (−0.0641 − 0.111i)10-s + (0.0694 − 0.259i)11-s + (0.183 + 0.983i)13-s + (0.339 − 0.233i)14-s − 0.520·16-s + 0.0111·17-s + (−0.193 + 0.0518i)19-s + (0.249 + 0.0669i)20-s + (−0.0552 − 0.0956i)22-s − 0.665i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95985 + 0.619012i\)
\(L(\frac12)\) \(\approx\) \(1.95985 + 0.619012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.483i)T \)
13 \( 1 + (-0.659 - 3.54i)T \)
good2 \( 1 + (-0.411 + 0.411i)T - 2iT^{2} \)
5 \( 1 + (-0.180 + 0.672i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.230 + 0.859i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 0.0460T + 17T^{2} \)
19 \( 1 + (0.843 - 0.225i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.19iT - 23T^{2} \)
29 \( 1 + (4.08 - 7.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.90 + 1.04i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.97 + 4.97i)T + 37iT^{2} \)
41 \( 1 + (-8.56 + 2.29i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.29 + 0.746i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-12.1 - 3.24i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.89 - 8.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.25 - 6.25i)T - 59iT^{2} \)
61 \( 1 + (-0.877 - 0.506i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.42 - 0.648i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.798 - 0.213i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.09 + 15.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.73 + 8.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.50 - 3.50i)T + 83iT^{2} \)
89 \( 1 + (-3.69 + 3.69i)T - 89iT^{2} \)
97 \( 1 + (0.288 - 1.07i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.82120191338954881641263683334, −8.997572330499636258102781620668, −8.887226010000870971699933729711, −7.75504791826990167513818591468, −7.05833046295088006236522543232, −5.76704930111700747076006617728, −4.69874998616149524514309352293, −4.05602368163807505702107118188, −2.75884132940561902223608779580, −1.60386442257045306505743895067, 1.05805616226933395218640104386, 2.39294332396832906180719843729, 3.96339290154788302006328207841, 4.92869896146895558621293974831, 5.67147110491883298765731761504, 6.58570367346839296579391859663, 7.52920590092175376487730442193, 8.307483285843874549219516935883, 9.450124591832689356097275757157, 10.26895040831491582857682797195

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