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Properties

Label 2-24e2-144.133-c1-0-2
Degree $2$
Conductor $576$
Sign $-0.903 - 0.428i$
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  + (−1.71 − 0.222i)3-s + (−0.798 + 2.97i)5-s + (1.78 + 1.02i)7-s + (2.90 + 0.764i)9-s + (0.446 − 0.119i)11-s + (−5.67 − 1.52i)13-s + (2.03 − 4.93i)15-s + 0.0443·17-s + (1.10 − 1.10i)19-s + (−2.83 − 2.16i)21-s + (−7.89 + 4.55i)23-s + (−3.90 − 2.25i)25-s + (−4.81 − 1.95i)27-s + (1.86 + 6.95i)29-s + (−0.542 − 0.939i)31-s + ⋯
L(s)  = 1  + (−0.991 − 0.128i)3-s + (−0.357 + 1.33i)5-s + (0.673 + 0.388i)7-s + (0.966 + 0.254i)9-s + (0.134 − 0.0360i)11-s + (−1.57 − 0.421i)13-s + (0.525 − 1.27i)15-s + 0.0107·17-s + (0.254 − 0.254i)19-s + (−0.617 − 0.472i)21-s + (−1.64 + 0.950i)23-s + (−0.781 − 0.451i)25-s + (−0.926 − 0.377i)27-s + (0.346 + 1.29i)29-s + (−0.0973 − 0.168i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.903 - 0.428i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.903 - 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113663 + 0.505493i\)
\(L(\frac12)\) \(\approx\) \(0.113663 + 0.505493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.222i)T \)
good5 \( 1 + (0.798 - 2.97i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.78 - 1.02i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.446 + 0.119i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (5.67 + 1.52i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 0.0443T + 17T^{2} \)
19 \( 1 + (-1.10 + 1.10i)T - 19iT^{2} \)
23 \( 1 + (7.89 - 4.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.86 - 6.95i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.542 + 0.939i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.769 + 0.769i)T + 37iT^{2} \)
41 \( 1 + (5.77 - 3.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (11.0 - 2.96i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.22 + 2.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \)
59 \( 1 + (-0.962 + 3.59i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.318 - 1.18i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.52 - 1.48i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.88iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 + (3.46 - 6.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.157 + 0.588i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 5.30iT - 89T^{2} \)
97 \( 1 + (5.88 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.23703754672421427486041476119, −10.28743672891078404431943956675, −9.774230881958587605515090276939, −8.165362935378210161269746766127, −7.33428195158756819267047777675, −6.71657480281835809467458086957, −5.57710997203509331008133997159, −4.76812365021875775140796546554, −3.36213236679137149268816431350, −1.99280758702198234251187224092, 0.32356862539353352251980286579, 1.82719783410628413636048220355, 4.15349820947213213454504602060, 4.68418889591703139535362072397, 5.47541583316775551294299008508, 6.70063080674411251486167382341, 7.72254229765154920567670243154, 8.487074722291197298065858788805, 9.719901383355346167518952189296, 10.22277671810259448164983803510

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