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Properties

Label 2-810-45.29-c2-0-44
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: $\chi_{810} (269, \cdot )$
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

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

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