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Properties

Label 2-24e2-144.85-c1-0-0
Degree $2$
Conductor $576$
Sign $-0.573 + 0.819i$
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  + (0.162 + 1.72i)3-s + (−2.21 + 0.592i)5-s + (−2.67 + 1.54i)7-s + (−2.94 + 0.559i)9-s + (0.918 − 3.42i)11-s + (−0.375 − 1.40i)13-s + (−1.38 − 3.71i)15-s − 1.69·17-s + (5.41 − 5.41i)19-s + (−3.09 − 4.35i)21-s + (−3.69 − 2.13i)23-s + (0.208 − 0.120i)25-s + (−1.44 − 4.99i)27-s + (−0.550 − 0.147i)29-s + (−3.59 + 6.22i)31-s + ⋯
L(s)  = 1  + (0.0936 + 0.995i)3-s + (−0.988 + 0.264i)5-s + (−1.00 + 0.582i)7-s + (−0.982 + 0.186i)9-s + (0.277 − 1.03i)11-s + (−0.104 − 0.389i)13-s + (−0.356 − 0.959i)15-s − 0.411·17-s + (1.24 − 1.24i)19-s + (−0.674 − 0.950i)21-s + (−0.771 − 0.445i)23-s + (0.0416 − 0.0240i)25-s + (−0.277 − 0.960i)27-s + (−0.102 − 0.0273i)29-s + (−0.645 + 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.573 + 0.819i$
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: \((2,\ 576,\ (\ :1/2),\ -0.573 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00154498 - 0.00296857i\)
\(L(\frac12)\) \(\approx\) \(0.00154498 - 0.00296857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.162 - 1.72i)T \)
good5 \( 1 + (2.21 - 0.592i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.67 - 1.54i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.918 + 3.42i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.375 + 1.40i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 1.69T + 17T^{2} \)
19 \( 1 + (-5.41 + 5.41i)T - 19iT^{2} \)
23 \( 1 + (3.69 + 2.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.550 + 0.147i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.59 - 6.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.59 - 2.59i)T + 37iT^{2} \)
41 \( 1 + (8.14 + 4.70i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.81 - 10.5i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.322 - 0.558i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.59 + 7.59i)T + 53iT^{2} \)
59 \( 1 + (5.82 - 1.55i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.88 + 1.04i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.07 - 4.02i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.81iT - 71T^{2} \)
73 \( 1 - 0.0254iT - 73T^{2} \)
79 \( 1 + (7.90 + 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.35 + 2.50i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.29iT - 89T^{2} \)
97 \( 1 + (-4.31 - 7.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.37288569095319122080062071424, −10.41367114005016429293113290417, −9.485805774706005835964099923121, −8.811975703415381434073253352648, −7.943736257845796848374996019624, −6.72438056468008920086938138460, −5.74828715067625767926584000127, −4.68539478611019522254993839610, −3.40991466491692255038221584902, −3.01468644319106500841304028001, 0.00181147838946709612760688392, 1.74022059696473557202734549313, 3.36740164991071048969575055290, 4.20939203441466289202886833491, 5.72391075771010502740327262845, 6.77576387590589427643002416459, 7.46207976397595429632398268366, 8.069405777574635416436403940778, 9.318975625380315183267351332557, 9.992449000741525032528693855663

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