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Properties

Label 2-819-91.20-c1-0-36
Degree $2$
Conductor $819$
Sign $0.697 + 0.716i$
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  + (2.47 − 0.664i)2-s + (3.97 − 2.29i)4-s + (0.281 − 0.281i)5-s + (−0.201 + 2.63i)7-s + (4.70 − 4.70i)8-s + (0.511 − 0.885i)10-s + (−0.939 − 3.50i)11-s + (3.44 − 1.06i)13-s + (1.25 + 6.67i)14-s + (3.95 − 6.84i)16-s + (−2.04 − 3.54i)17-s + (0.777 + 0.208i)19-s + (0.473 − 1.76i)20-s + (−4.66 − 8.07i)22-s + (4.41 + 2.54i)23-s + ⋯
L(s)  = 1  + (1.75 − 0.469i)2-s + (1.98 − 1.14i)4-s + (0.125 − 0.125i)5-s + (−0.0762 + 0.997i)7-s + (1.66 − 1.66i)8-s + (0.161 − 0.280i)10-s + (−0.283 − 1.05i)11-s + (0.955 − 0.294i)13-s + (0.334 + 1.78i)14-s + (0.987 − 1.71i)16-s + (−0.496 − 0.860i)17-s + (0.178 + 0.0478i)19-s + (0.105 − 0.395i)20-s + (−0.993 − 1.72i)22-s + (0.919 + 0.531i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.697 + 0.716i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.697 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.03653 - 1.70394i\)
\(L(\frac12)\) \(\approx\) \(4.03653 - 1.70394i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.201 - 2.63i)T \)
13 \( 1 + (-3.44 + 1.06i)T \)
good2 \( 1 + (-2.47 + 0.664i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.281 + 0.281i)T - 5iT^{2} \)
11 \( 1 + (0.939 + 3.50i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.04 + 3.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.777 - 0.208i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.41 - 2.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.00 - 1.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.44 - 4.44i)T - 31iT^{2} \)
37 \( 1 + (-0.463 - 1.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.578 + 2.15i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.99 + 5.99i)T + 47iT^{2} \)
53 \( 1 + 9.09T + 53T^{2} \)
59 \( 1 + (1.92 - 7.17i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.40 + 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.85 - 1.30i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.582 + 2.17i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.50 + 3.50i)T + 73iT^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 + (5.36 - 5.36i)T - 83iT^{2} \)
89 \( 1 + (-11.5 + 3.09i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.71 + 2.06i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.62921979905567596124344813338, −9.315060503775169828015855766102, −8.535668620236582846747700220014, −7.15633018220025286438854145313, −6.18074601555878853889183398716, −5.47657949571269928486148858993, −4.91956337648341181335950137288, −3.43627367754546229081571153054, −2.99472199715788109294079453760, −1.60301737128856079626106041578, 1.93164901169315097574866402660, 3.25731884028927156124797692410, 4.21652394142574516570888781613, 4.73647848501046277919726149577, 5.99920590606828671612873893483, 6.63769181294487679871496961267, 7.36749767091159857383656913156, 8.254706182208330109071804443871, 9.658152567491778424383778185775, 10.74407454802636365838019286145

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