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Properties

Label 2-810-45.14-c2-0-21
Degree $2$
Conductor $810$
Sign $0.522 - 0.852i$
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 + (−4.89 − 0.997i)5-s + (−0.704 + 0.406i)7-s − 2.82·8-s + (−2.24 − 6.70i)10-s + (13.3 − 7.68i)11-s + (5.10 + 2.94i)13-s + (−0.996 − 0.575i)14-s + (−2.00 − 3.46i)16-s − 12.8·17-s + 1.24·19-s + (6.62 − 7.48i)20-s + (18.8 + 10.8i)22-s + (2.39 − 4.15i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.979 − 0.199i)5-s + (−0.100 + 0.0581i)7-s − 0.353·8-s + (−0.224 − 0.670i)10-s + (1.21 − 0.698i)11-s + (0.392 + 0.226i)13-s + (−0.0711 − 0.0410i)14-s + (−0.125 − 0.216i)16-s − 0.758·17-s + 0.0654·19-s + (0.331 − 0.374i)20-s + (0.855 + 0.494i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.522 - 0.852i$
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: \((2,\ 810,\ (\ :1),\ 0.522 - 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.821462939\)
\(L(\frac12)\) \(\approx\) \(1.821462939\)
\(L(2)\) 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 \( 1 + (4.89 + 0.997i)T \)
good7 \( 1 + (0.704 - 0.406i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-13.3 + 7.68i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.10 - 2.94i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 12.8T + 289T^{2} \)
19 \( 1 - 1.24T + 361T^{2} \)
23 \( 1 + (-2.39 + 4.15i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-36.9 + 21.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (2.10 - 3.64i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 70.3iT - 1.36e3T^{2} \)
41 \( 1 + (6.09 + 3.52i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-35.5 + 20.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.8 - 69.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 63.7T + 2.80e3T^{2} \)
59 \( 1 + (-31.7 - 18.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41.4 - 71.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-77.0 - 44.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 69.6iT - 5.04e3T^{2} \)
73 \( 1 - 89.6iT - 5.32e3T^{2} \)
79 \( 1 + (67.0 + 116. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (54.5 + 94.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 137. iT - 7.92e3T^{2} \)
97 \( 1 + (-78.4 + 45.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.16288422018276912849846846285, −8.806163640978881807652917838130, −8.669184985757916299919219421863, −7.53438473828148498129463437883, −6.65766450585646688683355993329, −5.98209471227914911359373996108, −4.60993468227496691120267974576, −4.02335260674913528056436148356, −2.95164571100874786191989904165, −0.932099603229460782758466169489, 0.78493788701566893907115934662, 2.27139735458653082975685804872, 3.62847622748312572550362252916, 4.14936331237756148033578250081, 5.23097645726842980925037425562, 6.56665240917789391788096923166, 7.15045800524744369369433346501, 8.389071179396345288084563587347, 9.091510287179250376346346080968, 10.05936926156666934768444308888

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