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Properties

Label 2-819-91.23-c1-0-38
Degree $2$
Conductor $819$
Sign $-0.945 + 0.325i$
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  − 0.456i·2-s + 1.79·4-s + (−1.5 − 0.866i)5-s + (−2.29 + 1.32i)7-s − 1.73i·8-s + (−0.395 + 0.685i)10-s + (−3 − 1.73i)11-s + (−1 − 3.46i)13-s + (0.604 + 1.04i)14-s + 2.79·16-s − 17-s + (−4.58 + 2.64i)19-s + (−2.68 − 1.55i)20-s + (−0.791 + 1.37i)22-s − 0.582·23-s + ⋯
L(s)  = 1  − 0.323i·2-s + 0.895·4-s + (−0.670 − 0.387i)5-s + (−0.866 + 0.499i)7-s − 0.612i·8-s + (−0.125 + 0.216i)10-s + (−0.904 − 0.522i)11-s + (−0.277 − 0.960i)13-s + (0.161 + 0.279i)14-s + 0.697·16-s − 0.242·17-s + (−1.05 + 0.606i)19-s + (−0.600 − 0.346i)20-s + (−0.168 + 0.292i)22-s − 0.121·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.945 + 0.325i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114013 - 0.681374i\)
\(L(\frac12)\) \(\approx\) \(0.114013 - 0.681374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 - 1.32i)T \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + 0.456iT - 2T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + (4.58 - 2.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.582T + 23T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.708 + 0.409i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.55iT - 37T^{2} \)
41 \( 1 + (6.08 - 3.51i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.29 + 3.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.29 - 3.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.08 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.57iT - 59T^{2} \)
61 \( 1 + (6.58 + 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 3.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.87 + 5.70i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.29 + 9.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 15.5iT - 89T^{2} \)
97 \( 1 + (0.0825 + 0.0476i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.19971726657195593690834685080, −8.986997270069385768143460481821, −8.049799859398400018766564392834, −7.44320185302878018332866625172, −6.19457343705380935279880467336, −5.66881625047677998311853729916, −4.17894174282672868760479083984, −3.13224830595946569089129920730, −2.27164201374704330691778077303, −0.30147129766992763349773226462, 2.07322861226253045369013194778, 3.14235839686318368354568700506, 4.22412996180016135864895004713, 5.42878572671530588298447478557, 6.73436033304535145985452725206, 6.95075411061233516151617760267, 7.78884439329828323636685647560, 8.825192923574887293935389433681, 9.934870385051250067953196254230, 10.68104763119975624106694997004

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