↑

Properties

Label 2-24e2-36.31-c2-0-31
Degree $2$
Conductor $576$
Sign $-0.537 + 0.843i$
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  + (−0.456 − 2.96i)3-s + (−4.61 + 7.99i)5-s + (5.33 − 3.07i)7-s + (−8.58 + 2.70i)9-s + (−3.70 + 2.13i)11-s + (−0.869 + 1.50i)13-s + (25.8 + 10.0i)15-s + 12.3·17-s − 33.9i·19-s + (−11.5 − 14.4i)21-s + (3.35 + 1.93i)23-s + (−30.1 − 52.1i)25-s + (11.9 + 24.2i)27-s + (−17.8 − 30.9i)29-s + (−38.8 − 22.4i)31-s + ⋯
L(s)  = 1  + (−0.152 − 0.988i)3-s + (−0.923 + 1.59i)5-s + (0.761 − 0.439i)7-s + (−0.953 + 0.300i)9-s + (−0.336 + 0.194i)11-s + (−0.0668 + 0.115i)13-s + (1.72 + 0.669i)15-s + 0.726·17-s − 1.78i·19-s + (−0.550 − 0.685i)21-s + (0.145 + 0.0841i)23-s + (−1.20 − 2.08i)25-s + (0.442 + 0.896i)27-s + (−0.615 − 1.06i)29-s + (−1.25 − 0.723i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.537 + 0.843i$
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: \((2,\ 576,\ (\ :1),\ -0.537 + 0.843i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8137765939\)
\(L(\frac12)\) \(\approx\) \(0.8137765939\)
\(L(2)\) 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 \( 1 + (0.456 + 2.96i)T \)
good5 \( 1 + (4.61 - 7.99i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-5.33 + 3.07i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (3.70 - 2.13i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (0.869 - 1.50i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 12.3T + 289T^{2} \)
19 \( 1 + 33.9iT - 361T^{2} \)
23 \( 1 + (-3.35 - 1.93i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (17.8 + 30.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (38.8 + 22.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 32.7T + 1.36e3T^{2} \)
41 \( 1 + (-21.8 + 37.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-33.9 + 19.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (39.8 - 23.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 46.3T + 2.80e3T^{2} \)
59 \( 1 + (-23.2 - 13.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.4 + 40.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (56.9 + 32.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 96.7iT - 5.04e3T^{2} \)
73 \( 1 + 14.0T + 5.32e3T^{2} \)
79 \( 1 + (34.3 - 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-81.7 + 47.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 81.8T + 7.92e3T^{2} \)
97 \( 1 + (7.99 + 13.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−10.68171625507651101573389989224, −9.311564943229174709599573838113, −7.85692569852437889434689670382, −7.58886683550515023265030889834, −6.88179856760308875457358175630, −5.86201204137129715741829905265, −4.47217507196062856193765146582, −3.18362102831001857667142190815, −2.18970742349999653161005649477, −0.33428846596358325966499720663, 1.35465218354157368898792164683, 3.40726191405830629300061953256, 4.31669337927725286236557074033, 5.20825720473010582846856552869, 5.70583160027115488911176796554, 7.71459682740646301348227161330, 8.273012224663223106238725004019, 8.982199259599485770850450969343, 9.828026310393758868837030010071, 10.92560090548192279346756020291

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