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Properties

Label 2-819-91.33-c1-0-0
Degree $2$
Conductor $819$
Sign $-0.862 + 0.505i$
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  + (−0.323 + 1.20i)2-s + (0.376 + 0.217i)4-s + (−0.986 − 3.68i)5-s + (−1.80 − 1.93i)7-s + (−2.15 + 2.15i)8-s + 4.76·10-s + (−3.16 + 3.16i)11-s + (3.57 + 0.437i)13-s + (2.91 − 1.56i)14-s + (−1.47 − 2.54i)16-s + (−0.601 + 1.04i)17-s + (−3.79 + 3.79i)19-s + (0.428 − 1.59i)20-s + (−2.79 − 4.84i)22-s + (−2.92 + 1.69i)23-s + ⋯
L(s)  = 1  + (−0.228 + 0.854i)2-s + (0.188 + 0.108i)4-s + (−0.441 − 1.64i)5-s + (−0.683 − 0.729i)7-s + (−0.761 + 0.761i)8-s + 1.50·10-s + (−0.953 + 0.953i)11-s + (0.992 + 0.121i)13-s + (0.780 − 0.417i)14-s + (−0.367 − 0.637i)16-s + (−0.145 + 0.252i)17-s + (−0.870 + 0.870i)19-s + (0.0958 − 0.357i)20-s + (−0.596 − 1.03i)22-s + (−0.610 + 0.352i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.862 + 0.505i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00415373 - 0.0152952i\)
\(L(\frac12)\) \(\approx\) \(0.00415373 - 0.0152952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.93i)T \)
13 \( 1 + (-3.57 - 0.437i)T \)
good2 \( 1 + (0.323 - 1.20i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.986 + 3.68i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.16 - 3.16i)T - 11iT^{2} \)
17 \( 1 + (0.601 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.79 - 3.79i)T - 19iT^{2} \)
23 \( 1 + (2.92 - 1.69i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.96 + 6.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.16 + 1.38i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.75 + 1.80i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.914 + 3.41i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.74 - 5.04i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.98 - 1.33i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.14 - 1.98i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.88 + 1.57i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 5.40iT - 61T^{2} \)
67 \( 1 + (3.83 + 3.83i)T + 67iT^{2} \)
71 \( 1 + (3.05 - 11.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.92 + 7.16i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.76 + 9.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.00 + 5.00i)T - 83iT^{2} \)
89 \( 1 + (1.36 - 5.08i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.69 - 0.722i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.56362275404573670463433156678, −9.709973599966354669650521837180, −8.707859796205176175310671730641, −8.126717356405461971110213613809, −7.49683058078507616562469697791, −6.43463675182496743688677772185, −5.60262699737475566751861567136, −4.53810006313570712276553964523, −3.64535533136031957357908056152, −1.84669367938750404040255206133, 0.00764085789745423866288532417, 2.20079148805710694459996272757, 3.10329757358679640161378265209, 3.50836728057754885660796174737, 5.46377402275008012698378885107, 6.56701193456797983157324219644, 6.75899281796001663348984193925, 8.238872435496812379466235508204, 8.982829333431931228526055926113, 10.23668158316102804819469491992

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