Properties

Label 2-819-117.43-c1-0-34
Degree $2$
Conductor $819$
Sign $0.854 - 0.518i$
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  + (−1.57 + 0.907i)2-s + (1.36 − 1.06i)3-s + (0.646 − 1.11i)4-s + (−2.33 + 1.35i)5-s + (−1.18 + 2.91i)6-s i·7-s − 1.28i·8-s + (0.742 − 2.90i)9-s + (2.44 − 4.24i)10-s + (−3.69 + 2.13i)11-s + (−0.305 − 2.21i)12-s + (−3.59 − 0.323i)13-s + (0.907 + 1.57i)14-s + (−1.76 + 4.33i)15-s + (2.45 + 4.25i)16-s + (2.60 + 4.51i)17-s + ⋯
L(s)  = 1  + (−1.11 + 0.641i)2-s + (0.789 − 0.613i)3-s + (0.323 − 0.559i)4-s + (−1.04 + 0.603i)5-s + (−0.484 + 1.18i)6-s − 0.377i·7-s − 0.453i·8-s + (0.247 − 0.968i)9-s + (0.774 − 1.34i)10-s + (−1.11 + 0.642i)11-s + (−0.0881 − 0.640i)12-s + (−0.995 − 0.0897i)13-s + (0.242 + 0.419i)14-s + (−0.455 + 1.11i)15-s + (0.614 + 1.06i)16-s + (0.632 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.854 - 0.518i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.854 - 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788963 + 0.220683i\)
\(L(\frac12)\) \(\approx\) \(0.788963 + 0.220683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 + 1.06i)T \)
7 \( 1 + iT \)
13 \( 1 + (3.59 + 0.323i)T \)
good2 \( 1 + (1.57 - 0.907i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.33 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.69 - 2.13i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.60 - 4.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.12 + 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.45T + 23T^{2} \)
29 \( 1 + (-1.50 - 2.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.41 + 3.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.28 - 1.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.26iT - 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 + (7.77 + 4.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.31T + 53T^{2} \)
59 \( 1 + (4.41 + 2.54i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 2.07iT - 67T^{2} \)
71 \( 1 + (-3.50 + 2.02i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + (-3.04 + 5.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.8 - 7.99i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-13.4 - 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−9.983091785549154154856552504432, −9.371275225181447913862955335112, −8.249056399860127176318343211018, −7.74046460258879293475851340770, −7.25155885066871762586111802554, −6.67622642831358489437334170904, −4.98996680555446672876590948575, −3.61161841190804755495303663504, −2.73215362471663802897405816055, −0.866337777942563748659923551460, 0.813739538762373626071730634476, 2.62951152852322789243529660678, 3.22959752530395909990556939314, 4.85438954675495873318257436786, 5.26502909413077180663368397961, 7.52870387682642229132451715724, 7.81704904711745511125131049130, 8.626672365041162226546158360673, 9.363394581062001863655333103833, 9.939311722614663252764354637123

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