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Properties

Label 2-24e2-144.85-c1-0-11
Degree $2$
Conductor $576$
Sign $0.616 - 0.787i$
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  + (1.15 + 1.28i)3-s + (1.21 − 0.326i)5-s + (0.707 − 0.408i)7-s + (−0.324 + 2.98i)9-s + (−0.497 + 1.85i)11-s + (0.116 + 0.434i)13-s + (1.82 + 1.19i)15-s + 6.62·17-s + (−1.18 + 1.18i)19-s + (1.34 + 0.439i)21-s + (−2.66 − 1.53i)23-s + (−2.95 + 1.70i)25-s + (−4.22 + 3.03i)27-s + (8.65 + 2.31i)29-s + (4.61 − 7.99i)31-s + ⋯
L(s)  = 1  + (0.667 + 0.744i)3-s + (0.544 − 0.145i)5-s + (0.267 − 0.154i)7-s + (−0.108 + 0.994i)9-s + (−0.149 + 0.559i)11-s + (0.0322 + 0.120i)13-s + (0.471 + 0.307i)15-s + 1.60·17-s + (−0.271 + 0.271i)19-s + (0.293 + 0.0960i)21-s + (−0.555 − 0.320i)23-s + (−0.591 + 0.341i)25-s + (−0.812 + 0.583i)27-s + (1.60 + 0.430i)29-s + (0.828 − 1.43i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.616 - 0.787i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.616 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82973 + 0.891004i\)
\(L(\frac12)\) \(\approx\) \(1.82973 + 0.891004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 - 1.28i)T \)
good5 \( 1 + (-1.21 + 0.326i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.707 + 0.408i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.497 - 1.85i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.116 - 0.434i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
19 \( 1 + (1.18 - 1.18i)T - 19iT^{2} \)
23 \( 1 + (2.66 + 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.65 - 2.31i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.61 + 7.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 + (9.15 + 5.28i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.66 - 6.19i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.140 - 0.244i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.83 - 4.83i)T + 53iT^{2} \)
59 \( 1 + (-7.15 + 1.91i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.87 + 2.64i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.39 + 5.22i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.27iT - 71T^{2} \)
73 \( 1 + 4.92iT - 73T^{2} \)
79 \( 1 + (7.70 + 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.9 - 2.92i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.44iT - 89T^{2} \)
97 \( 1 + (4.46 + 7.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.40458898564964740722491888140, −10.06236210168188465688763137119, −9.238683660587145062785603295256, −8.205078497045875511491250860630, −7.58698341883940982559731918595, −6.16489800355232574717957399983, −5.14228492633563454919732790974, −4.24020931076137668383034719300, −3.06243289637134511912747700021, −1.78376646125863442947404725986, 1.26463373518992946978520883902, 2.59203955736658442667829903154, 3.56735399989035946006430155117, 5.18024650416616371890244046023, 6.16456496866342663401818939731, 6.99266316830622363413297701712, 8.214683774422018281523560518047, 8.471950315911170227186960341722, 9.840676853025623562549163448366, 10.30831393095803525272912492252

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