Properties

Label 2-24e2-144.11-c1-0-5
Degree $2$
Conductor $576$
Sign $-0.486 - 0.873i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.874 + 1.49i)3-s + (1.76 + 0.473i)5-s + (−1.40 + 2.43i)7-s + (−1.46 + 2.61i)9-s + (−5.79 + 1.55i)11-s + (−1.10 − 0.296i)13-s + (0.837 + 3.05i)15-s + 0.699i·17-s + (2.01 + 2.01i)19-s + (−4.87 + 0.0286i)21-s + (4.91 − 2.83i)23-s + (−1.43 − 0.827i)25-s + (−5.19 + 0.0916i)27-s + (5.41 − 1.45i)29-s + (5.15 − 2.97i)31-s + ⋯
L(s)  = 1  + (0.505 + 0.863i)3-s + (0.789 + 0.211i)5-s + (−0.531 + 0.920i)7-s + (−0.489 + 0.871i)9-s + (−1.74 + 0.468i)11-s + (−0.307 − 0.0823i)13-s + (0.216 + 0.788i)15-s + 0.169i·17-s + (0.462 + 0.462i)19-s + (−1.06 + 0.00624i)21-s + (1.02 − 0.591i)23-s + (−0.286 − 0.165i)25-s + (−0.999 + 0.0176i)27-s + (1.00 − 0.269i)29-s + (0.925 − 0.534i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.486 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775714 + 1.32033i\)
\(L(\frac12)\) \(\approx\) \(0.775714 + 1.32033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.874 - 1.49i)T \)
good5 \( 1 + (-1.76 - 0.473i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.40 - 2.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.79 - 1.55i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.10 + 0.296i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 0.699iT - 17T^{2} \)
19 \( 1 + (-2.01 - 2.01i)T + 19iT^{2} \)
23 \( 1 + (-4.91 + 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.41 + 1.45i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.15 + 2.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.48 - 3.48i)T + 37iT^{2} \)
41 \( 1 + (-1.55 - 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.90 - 7.09i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.45 - 4.45i)T - 53iT^{2} \)
59 \( 1 + (-3.65 + 13.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.54 - 13.2i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.45 - 12.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.21iT - 71T^{2} \)
73 \( 1 + 3.75iT - 73T^{2} \)
79 \( 1 + (-2.96 - 1.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.308 + 1.15i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 0.391T + 89T^{2} \)
97 \( 1 + (0.875 - 1.51i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.62888048110700467434178235130, −10.02575873429099305499797077419, −9.511812785445232761414086407381, −8.462291575141663701845281239392, −7.69891674912409922571469334869, −6.25053938917305861174781617690, −5.41031005943228073521475688718, −4.58142944692141029274044977123, −2.86229696505213584370219158927, −2.48125402236523393749951256974, 0.795759478636129241866377061360, 2.42309962068726563849042880791, 3.30886213275036688390678649706, 4.98097993202266562454924634587, 5.92540675339412676374085305077, 7.02165522527546184160121809278, 7.61977882655011124213042523105, 8.628227153537648439904469179023, 9.562480150448393227913284035160, 10.32459437649214748448654787975

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