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Properties

Label 2-810-9.4-c3-0-39
Degree $2$
Conductor $810$
Sign $0.939 + 0.342i$
Analytic cond. $47.7915$
Root an. cond. $6.91314$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−2.5 + 4.33i)5-s + (6.5 + 11.2i)7-s − 7.99·8-s − 10·10-s + (−15 − 25.9i)11-s + (30.5 − 52.8i)13-s + (−12.9 + 22.5i)14-s + (−8 − 13.8i)16-s − 12·17-s − 49·19-s + (−10 − 17.3i)20-s + (30 − 51.9i)22-s + (9 − 15.5i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.350 + 0.607i)7-s − 0.353·8-s − 0.316·10-s + (−0.411 − 0.712i)11-s + (0.650 − 1.12i)13-s + (−0.248 + 0.429i)14-s + (−0.125 − 0.216i)16-s − 0.171·17-s − 0.591·19-s + (−0.111 − 0.193i)20-s + (0.290 − 0.503i)22-s + (0.0815 − 0.141i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.722854726\)
\(L(\frac12)\) \(\approx\) \(1.722854726\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
5 \( 1 + (2.5 - 4.33i)T \)
good7 \( 1 + (-6.5 - 11.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (15 + 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-30.5 + 52.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 12T + 4.91e3T^{2} \)
19 \( 1 + 49T + 6.85e3T^{2} \)
23 \( 1 + (-9 + 15.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (93 + 161. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-80 + 138. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 91T + 5.06e4T^{2} \)
41 \( 1 + (-189 + 327. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-134 - 232. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-72 - 124. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 570T + 1.48e5T^{2} \)
59 \( 1 + (-102 + 176. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-438.5 - 759. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-93.5 + 161. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 606T + 3.57e5T^{2} \)
73 \( 1 - 431T + 3.89e5T^{2} \)
79 \( 1 + (575.5 + 996. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-51 - 88.3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 984T + 7.04e5T^{2} \)
97 \( 1 + (-132.5 - 229. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.753140660791585664913237596517, −8.594750303052224534446509784490, −8.145708006717283551159484393413, −7.27112202969919724163430326340, −6.03326426346443778690420261439, −5.68318051106950629583654788270, −4.44908330928257312304200325030, −3.39088723236665032727912807706, −2.37287680295988863705438851330, −0.44385291053414054271330296253, 1.14577623768893148799842494685, 2.13153649240579534833368947121, 3.59505545568236913066433889267, 4.42038654623191290303836902343, 5.12744425662125923459648681894, 6.42304786264502955777204880686, 7.28111580912440758886931503730, 8.348615628113532852587654507799, 9.146569516561081907129803724991, 10.01332270960099688817827819702

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