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Properties

Label 2-810-9.4-c3-0-41
Degree $2$
Conductor $810$
Sign $0.173 + 0.984i$
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 + (−7 − 12.1i)7-s − 7.99·8-s + 10·10-s + (−3 − 5.19i)11-s + (−34 + 58.8i)13-s + (14 − 24.2i)14-s + (−8 − 13.8i)16-s + 78·17-s + 44·19-s + (10 + 17.3i)20-s + (6 − 10.3i)22-s + (−60 + 103. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.377 − 0.654i)7-s − 0.353·8-s + 0.316·10-s + (−0.0822 − 0.142i)11-s + (−0.725 + 1.25i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + 1.11·17-s + 0.531·19-s + (0.111 + 0.193i)20-s + (0.0581 − 0.100i)22-s + (−0.543 + 0.942i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
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: \((2,\ 810,\ (\ :3/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.189582651\)
\(L(\frac12)\) \(\approx\) \(1.189582651\)
\(L(\frac{5}{2})\) 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 \( 1 + (7 + 12.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (34 - 58.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 78T + 4.91e3T^{2} \)
19 \( 1 - 44T + 6.85e3T^{2} \)
23 \( 1 + (60 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (63 + 109. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-122 + 211. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 304T + 5.06e4T^{2} \)
41 \( 1 + (-240 + 415. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (52 + 90.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (300 + 519. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 258T + 1.48e5T^{2} \)
59 \( 1 + (267 - 462. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (181 + 313. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-134 + 232. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 972T + 3.57e5T^{2} \)
73 \( 1 - 470T + 3.89e5T^{2} \)
79 \( 1 + (622 + 1.07e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (198 + 342. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 972T + 7.04e5T^{2} \)
97 \( 1 + (-23 - 39.8i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.654815656473822269402845699659, −8.821109965430760364129334379176, −7.66864431526888342118215882008, −7.18414038448278547527218492124, −6.10101231172062599908926530671, −5.30195790978287636700409965502, −4.26999220691637587166254516810, −3.42572469535070706395919132657, −1.88304406892896568762972992588, −0.28124779898681569155790539712, 1.30458702128427782373890446337, 2.79179759940288663010719156394, 3.17968935316439485258884916323, 4.72993058818170728910832964292, 5.53635420023510681536322881769, 6.34557156976400250152855756372, 7.51340396021370086516600660038, 8.401728481077574997744826076223, 9.545811076780990486209945250317, 10.07515962067977572533162886955

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