↑

Properties

Label 2-810-45.29-c2-0-43
Degree $2$
Conductor $810$
Sign $-0.819 + 0.573i$
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 + (4.82 + 1.29i)5-s + (−3.46 − 2i)7-s − 2.82·8-s + (5 − 5i)10-s + (−9.79 − 5.65i)11-s + (15.5 − 9i)13-s + (−4.89 + 2.82i)14-s + (−2.00 + 3.46i)16-s − 1.41·17-s − 24·19-s + (−2.58 − 9.65i)20-s + (−13.8 + 7.99i)22-s + (−19.7 − 34.2i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.965 + 0.258i)5-s + (−0.494 − 0.285i)7-s − 0.353·8-s + (0.5 − 0.5i)10-s + (−0.890 − 0.514i)11-s + (1.19 − 0.692i)13-s + (−0.349 + 0.202i)14-s + (−0.125 + 0.216i)16-s − 0.0831·17-s − 1.26·19-s + (−0.129 − 0.482i)20-s + (−0.629 + 0.363i)22-s + (−0.860 − 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.819 + 0.573i$
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: \((2,\ 810,\ (\ :1),\ -0.819 + 0.573i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.773865369\)
\(L(\frac12)\) \(\approx\) \(1.773865369\)
\(L(2)\) 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 \( 1 + (-4.82 - 1.29i)T \)
good7 \( 1 + (3.46 + 2i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (9.79 + 5.65i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-15.5 + 9i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 1.41T + 289T^{2} \)
19 \( 1 + 24T + 361T^{2} \)
23 \( 1 + (19.7 + 34.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-33.0 - 19.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 56iT - 1.36e3T^{2} \)
41 \( 1 + (-20.8 + 12.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (69.2 + 40i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-14.1 + 24.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 4.24T + 2.80e3T^{2} \)
59 \( 1 + (-53.8 + 31.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (55 - 95.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.7 + 16i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 + 46iT - 5.32e3T^{2} \)
79 \( 1 + (18 - 31.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (2.82 - 4.89i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 57.9iT - 7.92e3T^{2} \)
97 \( 1 + (12.1 + 7i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−10.24146782330513975591581350451, −8.848477037713125335135345049566, −8.324146383948011889259257555825, −6.78592066001880024306061910853, −6.08471170345148277458666718157, −5.32382082291371882921987154686, −4.05706293717724323049388026349, −3.01003296777598157755723121571, −2.07634648118881567244856126063, −0.50353358374894542628718747165, 1.69102391259065240485848442538, 2.92961511656016860760781449160, 4.25376720637205307909853297062, 5.16743854681604388496379550363, 6.24715118427881744671771112001, 6.47855142294435845755734508196, 7.906806348490024739860424750099, 8.598899650970177406763828353669, 9.544623602822187514047734805279, 10.14537176617785216042701664817

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