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Properties

Label 2-810-45.14-c2-0-8
Degree $2$
Conductor $810$
Sign $-0.930 + 0.366i$
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 + (3.31 + 3.74i)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 + (−9.79 + 1.99i)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 + (0.662 + 0.748i)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 + (−0.489 + 0.0997i)20-s + (−0.855 − 0.494i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.930 + 0.366i$
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: \((2,\ 810,\ (\ :1),\ -0.930 + 0.366i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.047064132\)
\(L(\frac12)\) \(\approx\) \(1.047064132\)
\(L(2)\) 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 \( 1 + (-3.31 - 3.74i)T \)
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

−10.56278028648636027595409259857, −9.644719276300861632386803953870, −8.839374195868927835822493049328, −7.50608927319200846947326510141, −7.27560470720653643265265593444, −6.08073331269603452922736376154, −5.35825582067984182343999032342, −4.39623028454586611680212126169, −3.03548737831584495105297620840, −2.09138761141746423621700220638, 0.28349056199335134225424521840, 1.81074646093997913882376712388, 2.76688406007992074523015923868, 4.10804013910753723116328626251, 5.16157884552744510501618558804, 5.65342936061646373282582786533, 6.83965098024756968272827542006, 8.139633129792236423988145556417, 8.797536169127615290978025921770, 9.765495116943966444907509203409

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