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Properties

Label 2-810-9.4-c1-0-8
Degree $2$
Conductor $810$
Sign $0.173 + 0.984i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1 − 1.73i)7-s + 0.999·8-s − 0.999·10-s + (3 + 5.19i)11-s + (2 − 3.46i)13-s + (−0.999 + 1.73i)14-s + (−0.5 − 0.866i)16-s + 6·17-s − 4·19-s + (0.499 + 0.866i)20-s + (3 − 5.19i)22-s + (−0.499 − 0.866i)25-s − 3.99·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.377 − 0.654i)7-s + 0.353·8-s − 0.316·10-s + (0.904 + 1.56i)11-s + (0.554 − 0.960i)13-s + (−0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s + 1.45·17-s − 0.917·19-s + (0.111 + 0.193i)20-s + (0.639 − 1.10i)22-s + (−0.0999 − 0.173i)25-s − 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01401 - 0.850856i\)
\(L(\frac12)\) \(\approx\) \(1.01401 - 0.850856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.837944499553383758539843272516, −9.658540813604197887667641515491, −8.402415882840891306126028617959, −7.65335883144755573330110114224, −6.70537700023267427877426874441, −5.59524614834380880238427272187, −4.34181425340831731099735933241, −3.63098807556780542047947789900, −2.16084771198751002724771160442, −0.889499523116505208540889510551, 1.30437365209906239285431688448, 2.98956133962382164005705994400, 4.02092703353873053253180476707, 5.52519369583528994507275714442, 6.17353219904620548744328356898, 6.77080489699818364909131095102, 8.002531630508317854625495950061, 8.819754535869861129961924466271, 9.314344382080980584165466130250, 10.32991006418012040065465054538

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