↑

Properties

Label 2-810-45.29-c2-0-37
Degree $2$
Conductor $810$
Sign $0.522 + 0.852i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (4.89 − 0.997i)5-s + (−0.704 − 0.406i)7-s + 2.82·8-s + (−2.24 + 6.70i)10-s + (−13.3 − 7.68i)11-s + (5.10 − 2.94i)13-s + (0.996 − 0.575i)14-s + (−2.00 + 3.46i)16-s + 12.8·17-s + 1.24·19-s + (−6.62 − 7.48i)20-s + (18.8 − 10.8i)22-s + (−2.39 − 4.15i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.979 − 0.199i)5-s + (−0.100 − 0.0581i)7-s + 0.353·8-s + (−0.224 + 0.670i)10-s + (−1.21 − 0.698i)11-s + (0.392 − 0.226i)13-s + (0.0711 − 0.0410i)14-s + (−0.125 + 0.216i)16-s + 0.758·17-s + 0.0654·19-s + (−0.331 − 0.374i)20-s + (0.855 − 0.494i)22-s + (−0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.318898709\)
\(L(\frac12)\) \(\approx\) \(1.318898709\)
\(L(2)\) 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 \( 1 + (-4.89 + 0.997i)T \)
good7 \( 1 + (0.704 + 0.406i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (13.3 + 7.68i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.10 + 2.94i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 12.8T + 289T^{2} \)
19 \( 1 - 1.24T + 361T^{2} \)
23 \( 1 + (2.39 + 4.15i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (36.9 + 21.3i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (2.10 + 3.64i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 70.3iT - 1.36e3T^{2} \)
41 \( 1 + (-6.09 + 3.52i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-35.5 - 20.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.8 - 69.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 63.7T + 2.80e3T^{2} \)
59 \( 1 + (31.7 - 18.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-41.4 + 71.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-77.0 + 44.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 69.6iT - 5.04e3T^{2} \)
73 \( 1 + 89.6iT - 5.32e3T^{2} \)
79 \( 1 + (67.0 - 116. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-54.5 + 94.4i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 137. iT - 7.92e3T^{2} \)
97 \( 1 + (-78.4 - 45.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.740505327923035704601737647511, −9.137237971902647694368998507510, −8.097676443608605278498481418848, −7.52596016436500402409183700147, −6.17011785065761003336280000605, −5.74373935941498249258407134479, −4.84949504833730460524233811884, −3.33064447244414224376842587832, −2.01197622597557313278541947674, −0.50378024964345190638090622435, 1.41637144894735404375619213629, 2.44843960571936123371808196261, 3.44681203753931436365592326374, 4.88941631499730295623134958185, 5.65885783335361822588204327294, 6.81897807172325263173414204284, 7.72258914170737856643606492921, 8.655856660645748610718530307149, 9.640381229401798671136155491071, 10.07909176297505710366337492165

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