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Properties

Label 2-810-15.14-c2-0-15
Degree $2$
Conductor $810$
Sign $0.963 + 0.268i$
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  − 1.41·2-s + 2.00·4-s + (−4.81 − 1.34i)5-s − 0.364i·7-s − 2.82·8-s + (6.81 + 1.89i)10-s + 9.61i·11-s − 7.65i·13-s + 0.515i·14-s + 4.00·16-s − 12.3·17-s + 5.36·19-s + (−9.63 − 2.68i)20-s − 13.5i·22-s − 12.0·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + (−0.963 − 0.268i)5-s − 0.0521i·7-s − 0.353·8-s + (0.681 + 0.189i)10-s + 0.873i·11-s − 0.588i·13-s + 0.0368i·14-s + 0.250·16-s − 0.725·17-s + 0.282·19-s + (−0.481 − 0.134i)20-s − 0.617i·22-s − 0.524·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.963 + 0.268i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.963 + 0.268i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8905147541\)
\(L(\frac12)\) \(\approx\) \(0.8905147541\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 + (4.81 + 1.34i)T \)
good7 \( 1 + 0.364iT - 49T^{2} \)
11 \( 1 - 9.61iT - 121T^{2} \)
13 \( 1 + 7.65iT - 169T^{2} \)
17 \( 1 + 12.3T + 289T^{2} \)
19 \( 1 - 5.36T + 361T^{2} \)
23 \( 1 + 12.0T + 529T^{2} \)
29 \( 1 - 31.3iT - 841T^{2} \)
31 \( 1 - 4.91T + 961T^{2} \)
37 \( 1 + 40.2iT - 1.36e3T^{2} \)
41 \( 1 + 63.2iT - 1.68e3T^{2} \)
43 \( 1 - 53.8iT - 1.84e3T^{2} \)
47 \( 1 - 28.2T + 2.20e3T^{2} \)
53 \( 1 - 41.6T + 2.80e3T^{2} \)
59 \( 1 + 112. iT - 3.48e3T^{2} \)
61 \( 1 - 45.9T + 3.72e3T^{2} \)
67 \( 1 - 46.1iT - 4.48e3T^{2} \)
71 \( 1 - 125. iT - 5.04e3T^{2} \)
73 \( 1 + 59.0iT - 5.32e3T^{2} \)
79 \( 1 - 50.6T + 6.24e3T^{2} \)
83 \( 1 - 59.0T + 6.88e3T^{2} \)
89 \( 1 + 8.93iT - 7.92e3T^{2} \)
97 \( 1 + 131. iT - 9.40e3T^{2} \)
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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

−10.00433218520014633976121625603, −9.045639977402576316928378917442, −8.378135466032618762128731607750, −7.44887569011114604951401925246, −6.96090057094943580044919270163, −5.61842026089643995677127383808, −4.52971081173757623862090648120, −3.50765686880035032216397601133, −2.15014129839931782775864551296, −0.62783339500460381851436039960, 0.71457163023749345986382556872, 2.38143319027306817054174680861, 3.53935702211152230833240216851, 4.53299330974654897441827393930, 5.95177209443477511770791493047, 6.77908946108311865540685969913, 7.66387661074710123263126721625, 8.413521243107292495745000588701, 9.075730589423895438276740369879, 10.13912656167006814291219660278

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