↑

Properties

Label 2-24e2-576.155-c1-0-24
Degree $2$
Conductor $576$
Sign $-0.184 - 0.982i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.14 + 0.833i)2-s + (−1.72 − 0.124i)3-s + (0.609 + 1.90i)4-s + (−0.876 − 2.58i)5-s + (−1.86 − 1.58i)6-s + (−0.217 + 0.167i)7-s + (−0.891 + 2.68i)8-s + (2.96 + 0.428i)9-s + (1.15 − 3.68i)10-s + (0.277 + 4.23i)11-s + (−0.816 − 3.36i)12-s + (−1.17 + 1.03i)13-s + (−0.388 + 0.00930i)14-s + (1.19 + 4.57i)15-s + (−3.25 + 2.32i)16-s + (4.84 + 4.84i)17-s + ⋯
L(s)  = 1  + (0.807 + 0.589i)2-s + (−0.997 − 0.0716i)3-s + (0.304 + 0.952i)4-s + (−0.392 − 1.15i)5-s + (−0.763 − 0.645i)6-s + (−0.0823 + 0.0631i)7-s + (−0.315 + 0.948i)8-s + (0.989 + 0.142i)9-s + (0.364 − 1.16i)10-s + (0.0837 + 1.27i)11-s + (−0.235 − 0.971i)12-s + (−0.327 + 0.286i)13-s + (−0.103 + 0.00248i)14-s + (0.308 + 1.18i)15-s + (−0.814 + 0.580i)16-s + (1.17 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904206 + 1.09000i\)
\(L(\frac12)\) \(\approx\) \(0.904206 + 1.09000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.833i)T \)
3 \( 1 + (1.72 + 0.124i)T \)
good5 \( 1 + (0.876 + 2.58i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (0.217 - 0.167i)T + (1.81 - 6.76i)T^{2} \)
11 \( 1 + (-0.277 - 4.23i)T + (-10.9 + 1.43i)T^{2} \)
13 \( 1 + (1.17 - 1.03i)T + (1.69 - 12.8i)T^{2} \)
17 \( 1 + (-4.84 - 4.84i)T + 17iT^{2} \)
19 \( 1 + (-0.997 - 5.01i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-4.20 - 3.22i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (1.57 + 3.19i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (3.23 + 5.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.32 + 0.860i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-3.91 + 5.10i)T + (-10.6 - 39.6i)T^{2} \)
43 \( 1 + (-0.623 - 9.51i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (-0.905 - 3.37i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.38 - 2.07i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-9.88 + 3.35i)T + (46.8 - 35.9i)T^{2} \)
61 \( 1 + (2.50 + 5.07i)T + (-37.1 + 48.3i)T^{2} \)
67 \( 1 + (-0.642 + 9.80i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (5.60 - 13.5i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.21 + 5.33i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.50 + 9.36i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (2.60 - 7.68i)T + (-65.8 - 50.5i)T^{2} \)
89 \( 1 + (-5.38 + 12.9i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (2.46 + 1.42i)T + (48.5 + 84.0i)T^{2} \)
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

−11.35720625254989278192883618205, −10.11722108474611814306984312798, −9.195856924565859234953015521097, −7.85361130435930530515394918437, −7.43142161316622140128539063103, −6.15457047315354362365525978551, −5.42186600426405615696425998671, −4.57297178595612583453532057218, −3.82025435931842065705176350512, −1.64865728581607410435277363535, 0.74196103654669089679381644859, 2.86247122057293062027309607188, 3.56539088969833844708265897236, 5.00881929717538579312776706468, 5.62859897596658712538632017796, 6.88317096788698012350395174120, 7.15570056856506580620623826219, 8.989041727088863277492628329276, 10.17328105842565589429615620100, 10.70909642607493204768531019923

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