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Properties

Label 2-24e2-72.11-c1-0-21
Degree $2$
Conductor $576$
Sign $-0.722 + 0.691i$
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  + (1.31 + 1.12i)3-s + (−1.21 − 2.09i)5-s + (−3.96 − 2.28i)7-s + (0.456 + 2.96i)9-s + (−3.73 − 2.15i)11-s + (−1.88 + 1.08i)13-s + (0.774 − 4.12i)15-s − 3.90i·17-s − 5.93·19-s + (−2.62 − 7.47i)21-s + (2.94 + 5.10i)23-s + (−0.437 + 0.757i)25-s + (−2.74 + 4.41i)27-s + (−0.776 + 1.34i)29-s + (−0.925 + 0.534i)31-s + ⋯
L(s)  = 1  + (0.758 + 0.651i)3-s + (−0.541 − 0.938i)5-s + (−1.49 − 0.864i)7-s + (0.152 + 0.988i)9-s + (−1.12 − 0.650i)11-s + (−0.522 + 0.301i)13-s + (0.199 − 1.06i)15-s − 0.947i·17-s − 1.36·19-s + (−0.573 − 1.63i)21-s + (0.614 + 1.06i)23-s + (−0.0874 + 0.151i)25-s + (−0.528 + 0.849i)27-s + (−0.144 + 0.249i)29-s + (−0.166 + 0.0959i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206374 - 0.514011i\)
\(L(\frac12)\) \(\approx\) \(0.206374 - 0.514011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 - 1.12i)T \)
good5 \( 1 + (1.21 + 2.09i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.96 + 2.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.73 + 2.15i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.88 - 1.08i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.90iT - 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
23 \( 1 + (-2.94 - 5.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.776 - 1.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.925 - 0.534i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.02iT - 37T^{2} \)
41 \( 1 + (-10.4 + 6.04i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.995 + 1.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.70 + 6.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.97T + 53T^{2} \)
59 \( 1 + (-0.294 + 0.169i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.33 + 4.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.71 + 2.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.32T + 71T^{2} \)
73 \( 1 + 2.37T + 73T^{2} \)
79 \( 1 + (5.38 + 3.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.40 + 3.11i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.90iT - 89T^{2} \)
97 \( 1 + (1.32 - 2.28i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.33092234397298191838934401338, −9.369386998480914093611489764482, −8.854027169877583368724650159049, −7.76940322088093662900946557482, −7.06381562360578328800732947166, −5.56960283044647482856161792235, −4.52270343726841368863945684915, −3.66781009983968425484123280703, −2.65174269219830855700816043373, −0.26171570765177070184153108918, 2.51784770298770344702439008460, 2.86852295717349979812930184490, 4.17885016380916045959896725464, 5.93984795079221679394917389821, 6.64983266084247060542575647799, 7.46464883992097973803056382813, 8.322790166953819026901950917082, 9.260171279124628855329138409366, 10.14885489414407523360374913084, 10.89738657673020304084440959497

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