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Properties

Label 2-819-117.43-c1-0-26
Degree $2$
Conductor $819$
Sign $0.101 - 0.994i$
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  + (−0.777 + 0.448i)2-s + (−1.73 + 0.0290i)3-s + (−0.596 + 1.03i)4-s + (−2.07 + 1.19i)5-s + (1.33 − 0.799i)6-s + i·7-s − 2.86i·8-s + (2.99 − 0.100i)9-s + (1.07 − 1.86i)10-s + (4.58 − 2.64i)11-s + (1.00 − 1.80i)12-s + (−1.88 − 3.07i)13-s + (−0.448 − 0.777i)14-s + (3.55 − 2.13i)15-s + (0.0931 + 0.161i)16-s + (1.51 + 2.61i)17-s + ⋯
L(s)  = 1  + (−0.549 + 0.317i)2-s + (−0.999 + 0.0167i)3-s + (−0.298 + 0.517i)4-s + (−0.927 + 0.535i)5-s + (0.544 − 0.326i)6-s + 0.377i·7-s − 1.01i·8-s + (0.999 − 0.0335i)9-s + (0.340 − 0.589i)10-s + (1.38 − 0.797i)11-s + (0.289 − 0.521i)12-s + (−0.523 − 0.852i)13-s + (−0.119 − 0.207i)14-s + (0.918 − 0.551i)15-s + (0.0232 + 0.0403i)16-s + (0.366 + 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.101 - 0.994i$
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: \((2,\ 819,\ (\ :1/2),\ 0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434279 + 0.392404i\)
\(L(\frac12)\) \(\approx\) \(0.434279 + 0.392404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0290i)T \)
7 \( 1 - iT \)
13 \( 1 + (1.88 + 3.07i)T \)
good2 \( 1 + (0.777 - 0.448i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (2.07 - 1.19i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.58 + 2.64i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.51 - 2.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.142 - 0.0823i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.13T + 23T^{2} \)
29 \( 1 + (2.40 + 4.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.36 - 0.789i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.18 + 2.41i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + 1.92T + 43T^{2} \)
47 \( 1 + (1.85 + 1.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + (-4.52 - 2.61i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 - 7.17iT - 67T^{2} \)
71 \( 1 + (-3.62 + 2.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.0iT - 73T^{2} \)
79 \( 1 + (1.67 - 2.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.6 + 6.70i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-14.0 - 8.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.55iT - 97T^{2} \)
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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

−10.45377330640987670836931483075, −9.528369969709610858662219813914, −8.673278136348052008147013323391, −7.79216105773807830844215233429, −7.04272517468187819615884139768, −6.32171878627901445005742720136, −5.18044953583082137386116605827, −3.97215571182471464210796698517, −3.26905611324973695794578558259, −0.876031009010412356244016829663, 0.62584715212864743960729945868, 1.69421608527911831882406853046, 3.90260345409745892206869229437, 4.70222098692200694547573599010, 5.36639193869717113447303941487, 6.88451913822523040192367980296, 7.18463002637535824162670736000, 8.625858878048995259050787880099, 9.330507247101413538093341436724, 9.983755927310740984803466436000

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