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Properties

Label 2-810-9.4-c1-0-3
Degree $2$
Conductor $810$
Sign $-0.939 - 0.342i$
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 + (0.5 + 0.866i)7-s − 0.999·8-s − 0.999·10-s + (−2.5 + 4.33i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 6·17-s + 5·19-s + (−0.499 − 0.866i)20-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s − 5·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s − 0.353·8-s − 0.316·10-s + (−0.693 + 1.20i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.45·17-s + 1.14·19-s + (−0.111 − 0.193i)20-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s − 0.980·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 810,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212205 + 1.20347i\)
\(L(\frac12)\) \(\approx\) \(0.212205 + 1.20347i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71923736381369870936756785331, −9.513796483003927778005201441316, −8.954891634162904688562474182502, −7.890922550572815638136302182455, −7.05870347507207686627421478004, −6.44494177846937143488466013796, −5.25552357429392814403947732813, −4.47428517403903172147137115732, −3.35472269097484007578400158635, −2.05743060299448637290510215638, 0.52064783038718887033954295515, 2.14426198828478723991062551941, 3.32018167511013197192716409628, 4.42946551313378878683673142185, 5.15907863547186272717023210263, 6.20066908148836744066907098762, 7.42561644171098739360614113734, 8.149809521902787185100827241656, 9.257339951327747561909550095393, 9.894136239434421581607890412244

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