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Properties

Label 2-819-91.74-c1-0-41
Degree $2$
Conductor $819$
Sign $-0.144 + 0.989i$
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  + 2.11·2-s + 2.47·4-s + (−1.28 − 2.22i)5-s + (−2.36 − 1.18i)7-s + 1.01·8-s + (−2.71 − 4.70i)10-s + (−0.879 − 1.52i)11-s + (−0.451 − 3.57i)13-s + (−5.01 − 2.50i)14-s − 2.81·16-s + 7.50·17-s + (1.80 − 3.11i)19-s + (−3.18 − 5.51i)20-s + (−1.86 − 3.22i)22-s + 0.900·23-s + ⋯
L(s)  = 1  + 1.49·2-s + 1.23·4-s + (−0.574 − 0.994i)5-s + (−0.894 − 0.446i)7-s + 0.358·8-s + (−0.859 − 1.48i)10-s + (−0.265 − 0.459i)11-s + (−0.125 − 0.992i)13-s + (−1.33 − 0.668i)14-s − 0.703·16-s + 1.81·17-s + (0.413 − 0.715i)19-s + (−0.711 − 1.23i)20-s + (−0.397 − 0.687i)22-s + 0.187·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.144 + 0.989i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.144 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60737 - 1.85959i\)
\(L(\frac12)\) \(\approx\) \(1.60737 - 1.85959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 + 1.18i)T \)
13 \( 1 + (0.451 + 3.57i)T \)
good2 \( 1 - 2.11T + 2T^{2} \)
5 \( 1 + (1.28 + 2.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.879 + 1.52i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.50T + 17T^{2} \)
19 \( 1 + (-1.80 + 3.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.900T + 23T^{2} \)
29 \( 1 + (1.25 - 2.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.67 + 2.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.63T + 37T^{2} \)
41 \( 1 + (2.47 - 4.28i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.35 - 9.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.03 - 6.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.57 + 4.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 + (-3.66 + 6.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.52 + 2.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.86 - 11.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.32 + 12.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.05 + 7.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.42T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (4.95 + 8.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.10544057964038042391759659922, −9.197610672616173808802257664667, −8.102029946876461765884575815979, −7.31070840776234587442478057295, −6.14250181959698183986847953980, −5.37375400626235193519223304691, −4.65714456118588541842977631930, −3.51896022163045980128105819436, −3.01253101946028495738284148745, −0.74241090848725569070366321086, 2.31879274903250605189990002072, 3.39894321027351959014784968853, 3.80427255623158626174433025898, 5.20325730951750949711737533065, 5.89455377424856061463724831363, 6.93657780190533602167759653172, 7.35344412858059452737705452343, 8.790585274314244455704937452309, 9.869054661691551447588844946010, 10.57854157048866709615193106576

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