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Properties

Label 2-819-91.16-c1-0-22
Degree $2$
Conductor $819$
Sign $0.686 + 0.726i$
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  − 2.20·2-s + 2.85·4-s + (−1.98 + 3.44i)5-s + (1.55 − 2.14i)7-s − 1.87·8-s + (4.37 − 7.58i)10-s + (2.61 − 4.53i)11-s + (0.0218 + 3.60i)13-s + (−3.42 + 4.71i)14-s − 1.56·16-s − 2.66·17-s + (−3.23 − 5.60i)19-s + (−5.67 + 9.82i)20-s + (−5.76 + 9.99i)22-s − 0.120·23-s + ⋯
L(s)  = 1  − 1.55·2-s + 1.42·4-s + (−0.888 + 1.53i)5-s + (0.587 − 0.809i)7-s − 0.664·8-s + (1.38 − 2.39i)10-s + (0.789 − 1.36i)11-s + (0.00607 + 0.999i)13-s + (−0.914 + 1.26i)14-s − 0.391·16-s − 0.646·17-s + (−0.742 − 1.28i)19-s + (−1.26 + 2.19i)20-s + (−1.23 + 2.13i)22-s − 0.0252·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.686 + 0.726i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.686 + 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463227 - 0.199613i\)
\(L(\frac12)\) \(\approx\) \(0.463227 - 0.199613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 2.14i)T \)
13 \( 1 + (-0.0218 - 3.60i)T \)
good2 \( 1 + 2.20T + 2T^{2} \)
5 \( 1 + (1.98 - 3.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.61 + 4.53i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.66T + 17T^{2} \)
19 \( 1 + (3.23 + 5.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.120T + 23T^{2} \)
29 \( 1 + (0.968 + 1.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.99 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + (4.53 + 7.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.749 + 1.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.08 + 5.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.337 + 0.585i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.22T + 59T^{2} \)
61 \( 1 + (-2.53 - 4.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.99 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.13 + 8.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.362 + 0.627i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.09 - 3.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 0.253T + 89T^{2} \)
97 \( 1 + (-6.46 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.32368683635504035047250112612, −9.087409665540056500873961681964, −8.506298306703693749025109237646, −7.65406745164295453507765453791, −6.69721021416464398559272448136, −6.65595460109800923871739562603, −4.45054947123937775094968426164, −3.48759117519836015055995979204, −2.16283735773547119963745849421, −0.50629919386508437415357327006, 1.10271216518653431390637312108, 2.11069262779724822545810085923, 4.15873240768776954590868263847, 4.87493913167656229115803752883, 6.15409696075083900434043729622, 7.47468847473881071416110343218, 8.134247850935647330301042274318, 8.532230698254181774232738116066, 9.390591706152215710984209026339, 9.950434323333121027301646355200

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