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Properties

Label 2-810-9.4-c3-0-47
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 + (−16 − 27.7i)7-s + 7.99·8-s − 10·10-s + (−30 − 51.9i)11-s + (17 − 29.4i)13-s + (−31.9 + 55.4i)14-s + (−8 − 13.8i)16-s − 42·17-s − 76·19-s + (10 + 17.3i)20-s + (−60 + 103. i)22-s + (−12.5 − 21.6i)25-s − 68·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.863 − 1.49i)7-s + 0.353·8-s − 0.316·10-s + (−0.822 − 1.42i)11-s + (0.362 − 0.628i)13-s + (−0.610 + 1.05i)14-s + (−0.125 − 0.216i)16-s − 0.599·17-s − 0.917·19-s + (0.111 + 0.193i)20-s + (−0.581 + 1.00i)22-s + (−0.100 − 0.173i)25-s − 0.512·26-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

\(L(2)\) \(\approx\) \(0.6139817137\)
\(L(\frac12)\) \(\approx\) \(0.6139817137\)
\(L(\frac{5}{2})\) 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 \( 1 + (16 + 27.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (30 + 51.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-17 + 29.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 42T + 4.91e3T^{2} \)
19 \( 1 + 76T + 6.85e3T^{2} \)
23 \( 1 + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-116 + 200. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 134T + 5.06e4T^{2} \)
41 \( 1 + (-117 + 202. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-206 - 356. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (180 + 311. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 222T + 1.48e5T^{2} \)
59 \( 1 + (-330 + 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-245 - 424. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (406 - 703. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 120T + 3.57e5T^{2} \)
73 \( 1 - 746T + 3.89e5T^{2} \)
79 \( 1 + (76 + 131. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (402 + 696. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 678T + 7.04e5T^{2} \)
97 \( 1 + (97 + 168. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.401769846263524878522839750285, −8.399973783551182880335121994530, −7.80186703663369311866083677071, −6.62892279325052645183199684201, −5.77051940168282604092884616273, −4.40747624410671021806066259051, −3.57733588135989024841331541893, −2.57594961570633295824788850277, −0.883960023705282812159706060518, −0.23041430623214924736688899239, 1.96230643117006595950821303605, 2.76691416567493316497571756869, 4.38247335487342175081173749235, 5.36432603269556976888974606126, 6.34638505553800595651834526157, 6.80570931644698634474431332887, 7.954022952602187373535460372752, 8.877808702958485500081408438070, 9.475256333991774605802000272634, 10.21360463447797102004617943187

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