Properties

Label 2-24e2-576.131-c1-0-32
Degree $2$
Conductor $576$
Sign $-0.283 - 0.959i$
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  + (1.40 + 0.130i)2-s + (−1.50 + 0.851i)3-s + (1.96 + 0.367i)4-s + (1.73 + 3.52i)5-s + (−2.23 + 1.00i)6-s + (−1.31 + 1.71i)7-s + (2.72 + 0.774i)8-s + (1.55 − 2.56i)9-s + (1.98 + 5.19i)10-s + (−1.86 − 2.12i)11-s + (−3.27 + 1.11i)12-s + (0.0127 + 0.195i)13-s + (−2.07 + 2.24i)14-s + (−5.62 − 3.84i)15-s + (3.72 + 1.44i)16-s + (−1.83 + 1.83i)17-s + ⋯
L(s)  = 1  + (0.995 + 0.0922i)2-s + (−0.870 + 0.491i)3-s + (0.982 + 0.183i)4-s + (0.777 + 1.57i)5-s + (−0.912 + 0.408i)6-s + (−0.497 + 0.648i)7-s + (0.961 + 0.273i)8-s + (0.517 − 0.855i)9-s + (0.628 + 1.64i)10-s + (−0.562 − 0.641i)11-s + (−0.946 + 0.322i)12-s + (0.00354 + 0.0541i)13-s + (−0.555 + 0.600i)14-s + (−1.45 − 0.991i)15-s + (0.932 + 0.361i)16-s + (−0.446 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.283 - 0.959i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.283 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29336 + 1.73074i\)
\(L(\frac12)\) \(\approx\) \(1.29336 + 1.73074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.130i)T \)
3 \( 1 + (1.50 - 0.851i)T \)
good5 \( 1 + (-1.73 - 3.52i)T + (-3.04 + 3.96i)T^{2} \)
7 \( 1 + (1.31 - 1.71i)T + (-1.81 - 6.76i)T^{2} \)
11 \( 1 + (1.86 + 2.12i)T + (-1.43 + 10.9i)T^{2} \)
13 \( 1 + (-0.0127 - 0.195i)T + (-12.8 + 1.69i)T^{2} \)
17 \( 1 + (1.83 - 1.83i)T - 17iT^{2} \)
19 \( 1 + (0.326 - 0.218i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (3.00 + 3.91i)T + (-5.95 + 22.2i)T^{2} \)
29 \( 1 + (-8.17 + 2.77i)T + (23.0 - 17.6i)T^{2} \)
31 \( 1 + (-3.06 + 5.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.93 - 2.90i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (1.44 - 1.10i)T + (10.6 - 39.6i)T^{2} \)
43 \( 1 + (-7.00 - 7.99i)T + (-5.61 + 42.6i)T^{2} \)
47 \( 1 + (2.12 - 7.94i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.19 + 11.0i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (-7.88 + 3.88i)T + (35.9 - 46.8i)T^{2} \)
61 \( 1 + (-10.2 + 3.46i)T + (48.3 - 37.1i)T^{2} \)
67 \( 1 + (-2.11 + 2.40i)T + (-8.74 - 66.4i)T^{2} \)
71 \( 1 + (7.34 - 3.04i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.73 + 1.13i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.67 + 13.6i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (1.12 - 2.27i)T + (-50.5 - 65.8i)T^{2} \)
89 \( 1 + (7.68 - 3.18i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-14.0 + 8.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.04894090306891891289282590727, −10.33614526648705015320965073487, −9.746501520681905279125363656975, −8.130701349025139233074162624927, −6.72790352057912363303632484043, −6.25711417832551202641036773142, −5.74722185116099248211823002195, −4.48373148979160013472378360843, −3.21475942186213023897826616459, −2.39824399703991149812536053029, 1.02203154075893294307074256838, 2.24778556940490738645252825348, 4.16187470810847333189593292545, 5.02837410049331332577277733936, 5.57349142599807131653344484887, 6.64195089435522399442756385979, 7.41592145843671940259266592745, 8.658079349529077219110839351853, 10.03968015839694001310801780127, 10.39382370418480863114981992799

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