↑

Properties

Label 2-24e2-36.31-c2-0-44
Degree $2$
Conductor $576$
Sign $-0.982 - 0.187i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
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  + (1.28 − 2.71i)3-s + (0.454 − 0.787i)5-s + (−6.10 + 3.52i)7-s + (−5.68 − 6.97i)9-s + (6.96 − 4.02i)11-s + (−3.35 + 5.81i)13-s + (−1.54 − 2.24i)15-s − 26.3·17-s − 20.5i·19-s + (1.69 + 21.0i)21-s + (−21.8 − 12.6i)23-s + (12.0 + 20.9i)25-s + (−26.2 + 6.44i)27-s + (−15.1 − 26.2i)29-s + (−0.120 − 0.0693i)31-s + ⋯
L(s)  = 1  + (0.428 − 0.903i)3-s + (0.0909 − 0.157i)5-s + (−0.872 + 0.503i)7-s + (−0.632 − 0.774i)9-s + (0.633 − 0.365i)11-s + (−0.258 + 0.447i)13-s + (−0.103 − 0.149i)15-s − 1.54·17-s − 1.08i·19-s + (0.0809 + 1.00i)21-s + (−0.949 − 0.547i)23-s + (0.483 + 0.837i)25-s + (−0.971 + 0.238i)27-s + (−0.523 − 0.906i)29-s + (−0.00387 − 0.00223i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.982 - 0.187i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ -0.982 - 0.187i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5773094583\)
\(L(\frac12)\) \(\approx\) \(0.5773094583\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.28 + 2.71i)T \)
good5 \( 1 + (-0.454 + 0.787i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (6.10 - 3.52i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-6.96 + 4.02i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.35 - 5.81i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 26.3T + 289T^{2} \)
19 \( 1 + 20.5iT - 361T^{2} \)
23 \( 1 + (21.8 + 12.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (15.1 + 26.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (0.120 + 0.0693i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 69.7T + 1.36e3T^{2} \)
41 \( 1 + (29.3 - 50.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.45 - 1.41i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-70.7 + 40.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 30.0T + 2.80e3T^{2} \)
59 \( 1 + (-77.1 - 44.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.0 + 41.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (44.0 + 25.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 68.4iT - 5.04e3T^{2} \)
73 \( 1 + 22.1T + 5.32e3T^{2} \)
79 \( 1 + (34.4 - 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-23.0 + 13.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 25.7T + 7.92e3T^{2} \)
97 \( 1 + (52.3 + 90.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−9.793754918040723013767022777221, −8.927308444577938364620104292836, −8.579080655642256098617341857716, −7.08039244687470828327739253797, −6.64195724904183915797220944581, −5.67884243367737673405103984922, −4.19284791989032045550605818813, −2.93687749708435053215462724053, −1.92206019955082242926572266944, −0.18989925551830900237166185814, 2.13092335600556559827842618812, 3.49021320816703705401388006855, 4.14046007063389863801903978635, 5.36814647983050308494856620997, 6.49653811766224687023341050395, 7.39473780179075446646157323039, 8.615628473730823239085039043239, 9.246789503109136490549068487450, 10.33756148342623260616295311061, 10.45318581206567798649721928000

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