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Properties

Label 2-810-9.7-c3-0-21
Degree $2$
Conductor $810$
Sign $0.342 + 0.939i$
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 + (−15.8 + 27.3i)7-s − 7.99·8-s − 10·10-s + (−26.3 + 45.6i)11-s + (−33.5 − 58.1i)13-s + (31.6 + 54.7i)14-s + (−8 + 13.8i)16-s − 19.4·17-s + 103.·19-s + (−10 + 17.3i)20-s + (52.6 + 91.2i)22-s + (−77.2 − 133. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.853 + 1.47i)7-s − 0.353·8-s − 0.316·10-s + (−0.721 + 1.25i)11-s + (−0.715 − 1.23i)13-s + (0.603 + 1.04i)14-s + (−0.125 + 0.216i)16-s − 0.277·17-s + 1.24·19-s + (−0.111 + 0.193i)20-s + (0.510 + 0.884i)22-s + (−0.699 − 1.21i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.482790226\)
\(L(\frac12)\) \(\approx\) \(1.482790226\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.8 - 27.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (26.3 - 45.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (33.5 + 58.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 19.4T + 4.91e3T^{2} \)
19 \( 1 - 103.T + 6.85e3T^{2} \)
23 \( 1 + (77.2 + 133. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-41.9 + 72.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-57.8 - 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 417.T + 5.06e4T^{2} \)
41 \( 1 + (-74.2 - 128. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-226. + 392. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-256. + 444. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 139.T + 1.48e5T^{2} \)
59 \( 1 + (-277. - 480. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-12.8 + 22.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-477. - 826. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 49.5T + 3.57e5T^{2} \)
73 \( 1 + 399.T + 3.89e5T^{2} \)
79 \( 1 + (375. - 650. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-587. + 1.01e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + (422. - 731. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.932038920404471654933564906676, −9.015317617998244321769053225069, −8.122882217597967854866921592851, −7.10388308575433575720723523820, −5.80365901738426656151275828149, −5.26995896378542345093673697644, −4.25403931384755717703032609722, −2.78430680165688868239427997136, −2.38896539304171023858898200699, −0.51523642695522175045970864992, 0.75512866105652672116826565436, 2.81443208887811426551863733765, 3.71033819405614995276575145561, 4.52435409345861721738027743379, 5.81025732892650599429158704454, 6.57742492263494001341656063450, 7.48698670015815005848981561688, 7.85543434236801148966399546574, 9.339875662497785678625031715898, 9.834552438591596388228364593626

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