Properties

Label 2-24e2-576.155-c1-0-81
Degree $2$
Conductor $576$
Sign $-0.985 + 0.170i$
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  + (−1.35 + 0.405i)2-s + (0.810 − 1.53i)3-s + (1.67 − 1.09i)4-s + (−0.190 − 0.560i)5-s + (−0.476 + 2.40i)6-s + (−1.38 + 1.05i)7-s + (−1.81 + 2.16i)8-s + (−1.68 − 2.48i)9-s + (0.485 + 0.682i)10-s + (−0.168 − 2.57i)11-s + (−0.328 − 3.44i)12-s + (−4.98 + 4.36i)13-s + (1.44 − 1.99i)14-s + (−1.01 − 0.162i)15-s + (1.58 − 3.67i)16-s + (−3.24 − 3.24i)17-s + ⋯
L(s)  = 1  + (−0.958 + 0.286i)2-s + (0.467 − 0.883i)3-s + (0.835 − 0.549i)4-s + (−0.0851 − 0.250i)5-s + (−0.194 + 0.980i)6-s + (−0.522 + 0.400i)7-s + (−0.643 + 0.765i)8-s + (−0.562 − 0.826i)9-s + (0.153 + 0.215i)10-s + (−0.0508 − 0.775i)11-s + (−0.0947 − 0.995i)12-s + (−1.38 + 1.21i)13-s + (0.385 − 0.533i)14-s + (−0.261 − 0.0420i)15-s + (0.396 − 0.917i)16-s + (−0.788 − 0.788i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.985 + 0.170i$
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: \((2,\ 576,\ (\ :1/2),\ -0.985 + 0.170i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0318202 - 0.370768i\)
\(L(\frac12)\) \(\approx\) \(0.0318202 - 0.370768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.405i)T \)
3 \( 1 + (-0.810 + 1.53i)T \)
good5 \( 1 + (0.190 + 0.560i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (1.38 - 1.05i)T + (1.81 - 6.76i)T^{2} \)
11 \( 1 + (0.168 + 2.57i)T + (-10.9 + 1.43i)T^{2} \)
13 \( 1 + (4.98 - 4.36i)T + (1.69 - 12.8i)T^{2} \)
17 \( 1 + (3.24 + 3.24i)T + 17iT^{2} \)
19 \( 1 + (-0.0330 - 0.166i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (5.65 + 4.33i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (1.27 + 2.57i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (-3.97 - 6.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.20 + 0.240i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.83 - 4.99i)T + (-10.6 - 39.6i)T^{2} \)
43 \( 1 + (0.331 + 5.06i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (-1.62 - 6.05i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.13 + 6.19i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-4.47 + 1.51i)T + (46.8 - 35.9i)T^{2} \)
61 \( 1 + (6.79 + 13.7i)T + (-37.1 + 48.3i)T^{2} \)
67 \( 1 + (0.155 - 2.37i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (-3.86 + 9.32i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.40 + 10.6i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.73 - 6.46i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.817 + 2.40i)T + (-65.8 - 50.5i)T^{2} \)
89 \( 1 + (2.37 - 5.74i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (9.57 + 5.53i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.978659910863361460587100881019, −9.202178572123882179395253905785, −8.593920725233855743887465230845, −7.75017373186158207894534849539, −6.70149255637663016412620631200, −6.32269018353008673023884538177, −4.83815871715643720839752270717, −2.92729865776983629246357957608, −2.02552118008884100286770826870, −0.24447931125828047232123340439, 2.21758126072619112717303571364, 3.22615958350927506811244773609, 4.28240461397616285981008735940, 5.66600295631716992524956456797, 7.05943907032748481855894431184, 7.72402264010299628989301586042, 8.633862023521677575779640585315, 9.696617992691314888547779395904, 10.06711232053987468829674549469, 10.70892745513432450730501860796

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