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Properties

Label 2-810-45.14-c2-0-22
Degree $2$
Conductor $810$
Sign $-0.453 - 0.891i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (4.41 + 2.34i)5-s + (−8.20 + 4.73i)7-s − 2.82·8-s + (0.246 + 7.06i)10-s + (17.6 − 10.1i)11-s + (5.29 + 3.05i)13-s + (−11.6 − 6.70i)14-s + (−2.00 − 3.46i)16-s + 24.0·17-s + 16.0·19-s + (−8.48 + 5.29i)20-s + (24.9 + 14.4i)22-s + (−21.6 + 37.4i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.882 + 0.469i)5-s + (−1.17 + 0.676i)7-s − 0.353·8-s + (0.0246 + 0.706i)10-s + (1.60 − 0.926i)11-s + (0.407 + 0.234i)13-s + (−0.828 − 0.478i)14-s + (−0.125 − 0.216i)16-s + 1.41·17-s + 0.846·19-s + (−0.424 + 0.264i)20-s + (1.13 + 0.655i)22-s + (−0.940 + 1.62i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.453 - 0.891i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ -0.453 - 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.423055987\)
\(L(\frac12)\) \(\approx\) \(2.423055987\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 \)
5 \( 1 + (-4.41 - 2.34i)T \)
good7 \( 1 + (8.20 - 4.73i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-17.6 + 10.1i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.29 - 3.05i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 24.0T + 289T^{2} \)
19 \( 1 - 16.0T + 361T^{2} \)
23 \( 1 + (21.6 - 37.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (16.9 - 9.81i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (15.2 - 26.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 44.8iT - 1.36e3T^{2} \)
41 \( 1 + (41.6 + 24.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (41.6 - 24.0i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-34.0 - 58.9i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 61.7T + 2.80e3T^{2} \)
59 \( 1 + (-43.6 - 25.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-24.4 - 42.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-47.7 - 27.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 23.5iT - 5.04e3T^{2} \)
73 \( 1 + 40.4iT - 5.32e3T^{2} \)
79 \( 1 + (16.6 + 28.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-11.3 - 19.6i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 79.5iT - 7.92e3T^{2} \)
97 \( 1 + (-103. + 59.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.00566955194929361846028636740, −9.368544545934877675018719905835, −8.864544073349622372005232215631, −7.51220970053560076470103503623, −6.64144903137933091593677927609, −5.86711798992777944457855213865, −5.53867896223070378068547320148, −3.57994839930169569398948437529, −3.27423469197452365632034654359, −1.48541710067391973338536609142, 0.78782236378520118299769936709, 1.88541770862977705734956279442, 3.34858627241559760335754089676, 4.09702150244491462751186373224, 5.25587196033240883501924104365, 6.29089985805214604774489244966, 6.83034533090962964661392403747, 8.239705235618937103267101547601, 9.383673036116932908189160627485, 9.883919900256570719453579137719

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