Properties

Label 2-24e2-576.517-c1-0-38
Degree $2$
Conductor $576$
Sign $-0.897 - 0.440i$
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  + (0.359 + 1.36i)2-s + (0.298 + 1.70i)3-s + (−1.74 + 0.983i)4-s + (0.924 − 1.05i)5-s + (−2.22 + 1.02i)6-s + (4.48 − 0.590i)7-s + (−1.97 − 2.02i)8-s + (−2.82 + 1.01i)9-s + (1.77 + 0.885i)10-s + (−1.71 + 3.47i)11-s + (−2.19 − 2.67i)12-s + (−0.777 + 2.29i)13-s + (2.41 + 5.91i)14-s + (2.07 + 1.26i)15-s + (2.06 − 3.42i)16-s + (−4.38 + 4.38i)17-s + ⋯
L(s)  = 1  + (0.254 + 0.967i)2-s + (0.172 + 0.985i)3-s + (−0.870 + 0.491i)4-s + (0.413 − 0.471i)5-s + (−0.908 + 0.417i)6-s + (1.69 − 0.223i)7-s + (−0.697 − 0.716i)8-s + (−0.940 + 0.339i)9-s + (0.560 + 0.279i)10-s + (−0.516 + 1.04i)11-s + (−0.634 − 0.772i)12-s + (−0.215 + 0.635i)13-s + (0.646 + 1.58i)14-s + (0.535 + 0.325i)15-s + (0.516 − 0.856i)16-s + (−1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.897 - 0.440i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.897 - 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401329 + 1.72831i\)
\(L(\frac12)\) \(\approx\) \(0.401329 + 1.72831i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.359 - 1.36i)T \)
3 \( 1 + (-0.298 - 1.70i)T \)
good5 \( 1 + (-0.924 + 1.05i)T + (-0.652 - 4.95i)T^{2} \)
7 \( 1 + (-4.48 + 0.590i)T + (6.76 - 1.81i)T^{2} \)
11 \( 1 + (1.71 - 3.47i)T + (-6.69 - 8.72i)T^{2} \)
13 \( 1 + (0.777 - 2.29i)T + (-10.3 - 7.91i)T^{2} \)
17 \( 1 + (4.38 - 4.38i)T - 17iT^{2} \)
19 \( 1 + (-0.960 - 4.82i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-1.02 + 7.76i)T + (-22.2 - 5.95i)T^{2} \)
29 \( 1 + (-7.06 - 0.462i)T + (28.7 + 3.78i)T^{2} \)
31 \( 1 + (0.129 + 0.0748i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.836 + 4.20i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-0.688 + 5.22i)T + (-39.6 - 10.6i)T^{2} \)
43 \( 1 + (0.680 + 0.335i)T + (26.1 + 34.1i)T^{2} \)
47 \( 1 + (6.96 + 1.86i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.43 - 3.64i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (3.61 - 4.12i)T + (-7.70 - 58.4i)T^{2} \)
61 \( 1 + (-0.273 + 4.16i)T + (-60.4 - 7.96i)T^{2} \)
67 \( 1 + (-14.0 + 6.93i)T + (40.7 - 53.1i)T^{2} \)
71 \( 1 + (-4.52 + 10.9i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.67 - 6.45i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.189 + 0.705i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-3.58 + 3.14i)T + (10.8 - 82.2i)T^{2} \)
89 \( 1 + (3.19 + 1.32i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-10.7 + 6.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.85721419910210005590939547965, −10.15456390077126965357433560565, −9.080452050489489480790534849558, −8.425858470541264766350620960543, −7.75620432773842942861192176152, −6.48903456006182589097606009786, −5.24270863822987432336313421977, −4.70392215165789355230529308465, −4.06046411996109193302923942504, −2.07394072227110035240241488094, 0.975532011311052411760550248048, 2.33035333463170034836823054263, 3.00102020581382412742436636557, 4.84163845145662668465344434099, 5.47387296278670663745029281414, 6.68985014250697775305492683597, 7.967148451399721563626588411280, 8.490506920500698085215427275060, 9.505010962141351365164208533328, 10.74674308030563951719332339625

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