Properties

Degree $2$
Conductor $810$
Sign $0.958 + 0.285i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.86 + 1.23i)5-s + (1.73 − i)7-s − 0.999i·8-s + (−1 + 2i)10-s + (1 + 1.73i)11-s + (5.19 + 3i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s − 2i·17-s + (0.133 + 2.23i)20-s + (1.73 + 0.999i)22-s + (3.46 + 2i)23-s + (1.96 − 4.59i)25-s + 6·26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.834 + 0.550i)5-s + (0.654 − 0.377i)7-s − 0.353i·8-s + (−0.316 + 0.632i)10-s + (0.301 + 0.522i)11-s + (1.44 + 0.832i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s − 0.485i·17-s + (0.0299 + 0.499i)20-s + (0.369 + 0.213i)22-s + (0.722 + 0.417i)23-s + (0.392 − 0.919i)25-s + 1.17·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.958 + 0.285i$
Motivic weight: \(1\)
Character: $\chi_{810} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17803 - 0.317062i\)
\(L(\frac12)\) \(\approx\) \(2.17803 - 0.317062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (1.86 - 1.23i)T \)
good7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.19 - 3i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.92 + 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (6.92 - 4i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.56202302456965840866854773761, −9.442052668288210158176849968996, −8.479814686167337644869900586287, −7.46023760517258651988001709602, −6.79068810139700082072572490684, −5.76300972825635462719417179980, −4.40049003068268393968876368221, −4.00562393910934380269582465942, −2.75971511172415920091243323121, −1.28756888547516624247362115688, 1.21192733268552206585795223702, 3.07480577387326517516897501884, 3.96175391746545540882408438466, 4.93710568150025938909243936881, 5.77469119071027812804284212739, 6.70927112430579125143091704597, 7.925249820418231375905185840640, 8.391558278783829664601080376607, 9.051976876232071449474348386038, 10.67888780693220233853450791457

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