Properties

Label 2-24e2-576.133-c1-0-73
Degree $2$
Conductor $576$
Sign $-0.515 + 0.856i$
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  + (1.23 − 0.682i)2-s + (−1.11 + 1.32i)3-s + (1.06 − 1.69i)4-s + (−0.339 − 1.00i)5-s + (−0.468 + 2.40i)6-s + (−1.07 − 1.39i)7-s + (0.171 − 2.82i)8-s + (−0.535 − 2.95i)9-s + (−1.10 − 1.00i)10-s + (−4.00 + 0.262i)11-s + (1.06 + 3.29i)12-s + (−1.18 − 1.35i)13-s + (−2.27 − 0.998i)14-s + (1.70 + 0.659i)15-s + (−1.71 − 3.61i)16-s + (1.11 − 1.11i)17-s + ⋯
L(s)  = 1  + (0.875 − 0.482i)2-s + (−0.640 + 0.767i)3-s + (0.534 − 0.845i)4-s + (−0.152 − 0.447i)5-s + (−0.191 + 0.981i)6-s + (−0.404 − 0.527i)7-s + (0.0606 − 0.998i)8-s + (−0.178 − 0.983i)9-s + (−0.349 − 0.318i)10-s + (−1.20 + 0.0791i)11-s + (0.306 + 0.952i)12-s + (−0.329 − 0.375i)13-s + (−0.609 − 0.266i)14-s + (0.441 + 0.170i)15-s + (−0.428 − 0.903i)16-s + (0.270 − 0.270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666378 - 1.17897i\)
\(L(\frac12)\) \(\approx\) \(0.666378 - 1.17897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 + 0.682i)T \)
3 \( 1 + (1.11 - 1.32i)T \)
good5 \( 1 + (0.339 + 1.00i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (1.07 + 1.39i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (4.00 - 0.262i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (1.18 + 1.35i)T + (-1.69 + 12.8i)T^{2} \)
17 \( 1 + (-1.11 + 1.11i)T - 17iT^{2} \)
19 \( 1 + (0.973 + 4.89i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-5.10 - 3.92i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (1.73 + 3.52i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (6.48 - 3.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0503 - 0.253i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-7.89 - 6.06i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (0.225 + 3.43i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (1.28 + 4.80i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.60 + 6.89i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-2.01 - 5.94i)T + (-46.8 + 35.9i)T^{2} \)
61 \( 1 + (0.298 - 0.147i)T + (37.1 - 48.3i)T^{2} \)
67 \( 1 + (-0.0366 + 0.559i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (3.49 - 8.44i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.585 - 1.41i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-8.83 + 2.36i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-15.8 - 5.37i)T + (65.8 + 50.5i)T^{2} \)
89 \( 1 + (-11.2 - 4.64i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-0.496 - 0.286i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63971045070951402396379105392, −9.868726763014981008655985492708, −9.046201278859005624560329790113, −7.49342205269538539770183794752, −6.56935841874923066975043386569, −5.27806619302536256780039531156, −4.97429806564669549323715082654, −3.79420733203272954263265346465, −2.76848953508130167285128278502, −0.59631468956316478369334183319, 2.19954769659510676200438563193, 3.23258668030539151743492002567, 4.76906927033415535448883971987, 5.65617669376577890225223911594, 6.33904652061406811072187892230, 7.34836088419595535667845493117, 7.86067768159801913254217822719, 9.062132225038543052908119236855, 10.69035514766259879850588088361, 11.01553949090744738199836130691

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