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Properties

Label 2-810-45.29-c2-0-8
Degree $2$
Conductor $810$
Sign $-0.991 + 0.131i$
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 + (4.43 + 2.30i)5-s + (6.62 + 3.82i)7-s + 2.82·8-s + (−5.96 + 3.80i)10-s + (−9.85 − 5.68i)11-s + (−12.4 + 7.19i)13-s + (−9.36 + 5.40i)14-s + (−2.00 + 3.46i)16-s − 17.3·17-s − 31.8·19-s + (−0.445 − 9.99i)20-s + (13.9 − 8.04i)22-s + (−8.25 − 14.3i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.887 + 0.460i)5-s + (0.946 + 0.546i)7-s + 0.353·8-s + (−0.596 + 0.380i)10-s + (−0.895 − 0.517i)11-s + (−0.958 + 0.553i)13-s + (−0.668 + 0.386i)14-s + (−0.125 + 0.216i)16-s − 1.01·17-s − 1.67·19-s + (−0.0222 − 0.499i)20-s + (0.633 − 0.365i)22-s + (−0.359 − 0.621i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.991 + 0.131i$
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: \((2,\ 810,\ (\ :1),\ -0.991 + 0.131i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7581223629\)
\(L(\frac12)\) \(\approx\) \(0.7581223629\)
\(L(2)\) 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 \( 1 + (-4.43 - 2.30i)T \)
good7 \( 1 + (-6.62 - 3.82i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (9.85 + 5.68i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (12.4 - 7.19i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 17.3T + 289T^{2} \)
19 \( 1 + 31.8T + 361T^{2} \)
23 \( 1 + (8.25 + 14.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-7.35 - 4.24i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-22.9 - 39.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 31.8iT - 1.36e3T^{2} \)
41 \( 1 + (59.2 - 34.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-27.1 - 15.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-23.0 + 39.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 53.6T + 2.80e3T^{2} \)
59 \( 1 + (56.0 - 32.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-48.1 + 83.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.49 + 0.862i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 26.5iT - 5.04e3T^{2} \)
73 \( 1 - 51.4iT - 5.32e3T^{2} \)
79 \( 1 + (29.6 - 51.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (3.46 - 6.00i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 115. iT - 7.92e3T^{2} \)
97 \( 1 + (-137. - 79.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.48499212255398245063828385907, −9.583842264106033958593254444241, −8.560962863846405475583141202398, −8.196815557169410226436249369138, −6.83434441973733073169702219820, −6.35146228269386786928788671210, −5.18877631279786098898015123028, −4.61492377247325666441855089994, −2.65686903664226566389111124205, −1.83568918939008415056922872126, 0.25885159211400976581743355970, 1.86016837664966719371811474965, 2.48638838736580409186474756258, 4.31195394088020805618199521790, 4.85402391665243348549599266895, 5.97896644749280585660022618280, 7.27000677946142866600806579214, 8.044078104642035740060390533687, 8.830338869055189531889682824629, 9.790197397640726199850835236540

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