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Properties

Label 2-24e2-9.7-c1-0-16
Degree $2$
Conductor $576$
Sign $-0.350 + 0.936i$
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 + 1.41i)3-s + (−0.5 − 0.866i)5-s + (0.724 − 1.25i)7-s + (−1.00 − 2.82i)9-s + (−1.72 + 2.98i)11-s + (−1.94 − 3.37i)13-s + (1.72 + 0.158i)15-s − 4.89·17-s − 4·19-s + (1.05 + 2.28i)21-s + (0.275 + 0.476i)23-s + (2 − 3.46i)25-s + (5.00 + 1.41i)27-s + (4.94 − 8.57i)29-s + (−3.72 − 6.45i)31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s + (−0.223 − 0.387i)5-s + (0.273 − 0.474i)7-s + (−0.333 − 0.942i)9-s + (−0.520 + 0.900i)11-s + (−0.540 − 0.936i)13-s + (0.445 + 0.0410i)15-s − 1.18·17-s − 0.917·19-s + (0.229 + 0.497i)21-s + (0.0573 + 0.0994i)23-s + (0.400 − 0.692i)25-s + (0.962 + 0.272i)27-s + (0.919 − 1.59i)29-s + (−0.668 − 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.268005 - 0.386535i\)
\(L(\frac12)\) \(\approx\) \(0.268005 - 0.386535i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.41i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.724 + 1.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.72 - 2.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.94 + 3.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-0.275 - 0.476i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.94 + 8.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.72 + 6.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.89T + 37T^{2} \)
41 \( 1 + (-1.05 - 1.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.17 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.17 - 7.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.898T + 53T^{2} \)
59 \( 1 + (0.174 + 0.301i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.949 + 1.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.17 + 2.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + (4.27 - 7.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.72 - 4.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + (2.94 - 5.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.47876343504928953716636541542, −9.794794286522626185926083432016, −8.770698254607986952513464468418, −7.84384654109947203945265689030, −6.79196902073354319482678750417, −5.67792344196394341805743545864, −4.62983801971823035040113728822, −4.16021212456707605471521812425, −2.47293557828317056910520970069, −0.27164024523269784222340370229, 1.79590261645023774044436844935, 2.98594603016891087920868813104, 4.65432499601786570478946862812, 5.53548095538216404492859524390, 6.69501396869844011243695972227, 7.11732569612177249784188057910, 8.451853873308789095398499946454, 8.898705811167742626945685369476, 10.56856485131131733105363901435, 10.97813553370871998394931979290

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