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Properties

Label 2-810-45.14-c2-0-6
Degree $2$
Conductor $810$
Sign $-0.536 + 0.843i$
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 + (0.222 + 4.99i)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 + (−8.87 − 4.60i)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 + (0.0445 + 0.999i)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 + (−0.443 − 0.230i)20-s + (−0.633 − 0.365i)22-s + (0.359 − 0.621i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.536 + 0.843i$
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: \((2,\ 810,\ (\ :1),\ -0.536 + 0.843i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7313207761\)
\(L(\frac12)\) \(\approx\) \(0.7313207761\)
\(L(2)\) 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 \( 1 + (-0.222 - 4.99i)T \)
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

−10.38394516010873009556349839190, −9.923641586163913786343345626352, −8.723929062415089709220994898584, −8.007347056854309159533860193315, −6.84684385564001976029306582066, −6.39713938823426912301596557355, −5.56669560709166364648959787253, −4.27031802973114169582677856874, −3.24214022101598285294608101211, −2.30515594382456853739764551768, 0.21881033034036597208565344524, 1.39256614680397711677424475081, 3.02625895712458743742057858478, 3.79449731758377353589193576026, 4.94836187998768466364343486840, 5.76165119099272916971245957783, 6.66965542556492304551971078085, 8.095109077431910032728197438056, 8.618425457151980616892584065593, 9.705251927187979710405938694569

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