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Properties

Label 2-819-91.45-c1-0-33
Degree $2$
Conductor $819$
Sign $0.187 + 0.982i$
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  + (−0.556 + 0.556i)2-s + 1.38i·4-s + (−0.542 + 2.02i)5-s + (−0.405 − 2.61i)7-s + (−1.88 − 1.88i)8-s + (−0.824 − 1.42i)10-s + (0.632 − 2.36i)11-s + (1.96 − 3.02i)13-s + (1.67 + 1.22i)14-s − 0.671·16-s − 6.55·17-s + (−7.74 + 2.07i)19-s + (−2.79 − 0.750i)20-s + (0.961 + 1.66i)22-s − 3.84i·23-s + ⋯
L(s)  = 1  + (−0.393 + 0.393i)2-s + 0.690i·4-s + (−0.242 + 0.906i)5-s + (−0.153 − 0.988i)7-s + (−0.664 − 0.664i)8-s + (−0.260 − 0.451i)10-s + (0.190 − 0.712i)11-s + (0.546 − 0.837i)13-s + (0.448 + 0.328i)14-s − 0.167·16-s − 1.58·17-s + (−1.77 + 0.475i)19-s + (−0.626 − 0.167i)20-s + (0.204 + 0.354i)22-s − 0.801i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.187 + 0.982i$
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: \((2,\ 819,\ (\ :1/2),\ 0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337631 - 0.279416i\)
\(L(\frac12)\) \(\approx\) \(0.337631 - 0.279416i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.405 + 2.61i)T \)
13 \( 1 + (-1.96 + 3.02i)T \)
good2 \( 1 + (0.556 - 0.556i)T - 2iT^{2} \)
5 \( 1 + (0.542 - 2.02i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.632 + 2.36i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 6.55T + 17T^{2} \)
19 \( 1 + (7.74 - 2.07i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.84iT - 23T^{2} \)
29 \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.457 - 0.122i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.00 + 5.00i)T + 37iT^{2} \)
41 \( 1 + (-11.0 + 2.94i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.810 + 0.467i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.03 + 1.88i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.08 + 1.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.92 + 3.92i)T - 59iT^{2} \)
61 \( 1 + (-8.13 - 4.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.1 + 2.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (9.44 + 2.52i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.62 - 9.79i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.07 - 1.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.52 + 1.52i)T + 83iT^{2} \)
89 \( 1 + (4.45 - 4.45i)T - 89iT^{2} \)
97 \( 1 + (-4.17 + 15.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.22687303571687819585246275428, −8.851929100738334690093723360224, −8.389913802912199019563014041858, −7.42958838719034791234039076234, −6.68648661077656673367794115026, −6.14197982947705295598377431658, −4.28088236127297770226378856822, −3.65005096762969608520320867701, −2.55405665369855860310841683112, −0.23890448004513370756885447174, 1.58243415775290256654055104960, 2.46040470664515385421791286753, 4.31224072512078599707152099894, 4.93193539664204141428695627664, 6.13398272267804425906539033611, 6.74706279400975246132335170471, 8.354865299320098029006048988794, 8.999257108960278113764452302387, 9.249879872282012494981243555292, 10.42676689309280855583082356261

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