Properties

Label 2-24e2-576.13-c1-0-18
Degree $2$
Conductor $576$
Sign $-0.976 - 0.213i$
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  + (−1.15 + 0.819i)2-s + (1.33 + 1.10i)3-s + (0.656 − 1.88i)4-s + (0.128 − 0.378i)5-s + (−2.44 − 0.179i)6-s + (−1.38 + 1.80i)7-s + (0.791 + 2.71i)8-s + (0.560 + 2.94i)9-s + (0.162 + 0.542i)10-s + (−5.67 − 0.371i)11-s + (2.96 − 1.79i)12-s + (−1.49 + 1.70i)13-s + (0.116 − 3.21i)14-s + (0.590 − 0.363i)15-s + (−3.13 − 2.48i)16-s + (0.935 + 0.935i)17-s + ⋯
L(s)  = 1  + (−0.814 + 0.579i)2-s + (0.770 + 0.637i)3-s + (0.328 − 0.944i)4-s + (0.0575 − 0.169i)5-s + (−0.997 − 0.0731i)6-s + (−0.524 + 0.683i)7-s + (0.279 + 0.960i)8-s + (0.186 + 0.982i)9-s + (0.0513 + 0.171i)10-s + (−1.71 − 0.112i)11-s + (0.855 − 0.518i)12-s + (−0.414 + 0.472i)13-s + (0.0312 − 0.860i)14-s + (0.152 − 0.0938i)15-s + (−0.784 − 0.620i)16-s + (0.227 + 0.227i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.976 - 0.213i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.976 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0820516 + 0.759112i\)
\(L(\frac12)\) \(\approx\) \(0.0820516 + 0.759112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.819i)T \)
3 \( 1 + (-1.33 - 1.10i)T \)
good5 \( 1 + (-0.128 + 0.378i)T + (-3.96 - 3.04i)T^{2} \)
7 \( 1 + (1.38 - 1.80i)T + (-1.81 - 6.76i)T^{2} \)
11 \( 1 + (5.67 + 0.371i)T + (10.9 + 1.43i)T^{2} \)
13 \( 1 + (1.49 - 1.70i)T + (-1.69 - 12.8i)T^{2} \)
17 \( 1 + (-0.935 - 0.935i)T + 17iT^{2} \)
19 \( 1 + (-0.719 + 3.61i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (4.47 - 3.43i)T + (5.95 - 22.2i)T^{2} \)
29 \( 1 + (3.89 - 7.89i)T + (-17.6 - 23.0i)T^{2} \)
31 \( 1 + (-3.85 - 2.22i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.930 - 4.67i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (5.12 - 3.93i)T + (10.6 - 39.6i)T^{2} \)
43 \( 1 + (-0.767 + 11.7i)T + (-42.6 - 5.61i)T^{2} \)
47 \( 1 + (-1.78 + 6.67i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.873 + 1.30i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-1.25 + 3.70i)T + (-46.8 - 35.9i)T^{2} \)
61 \( 1 + (-3.00 - 1.48i)T + (37.1 + 48.3i)T^{2} \)
67 \( 1 + (-0.404 - 6.16i)T + (-66.4 + 8.74i)T^{2} \)
71 \( 1 + (-1.44 - 3.48i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.52 + 13.3i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (11.6 + 3.11i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-7.52 + 2.55i)T + (65.8 - 50.5i)T^{2} \)
89 \( 1 + (2.28 - 0.947i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-8.69 + 5.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.62237840196023687405654136481, −10.10792689373401723343231324289, −9.215031460163719382817215872508, −8.640503443046645459236624338421, −7.77349379061973168971159694617, −6.91791439028409579318834173815, −5.47682995101148789823663670675, −4.97587322993168103774795963461, −3.16151318835687577361550443933, −2.11773354302224468399882435275, 0.47567437115031629320774796549, 2.27521747927144876326845260313, 3.01387944059369023202683019719, 4.21240453513638046162961501702, 6.02650355963319582816488631483, 7.15835041968655297983915990975, 7.86942830527286485119642435434, 8.311985187792699539532417002516, 9.794473800711592673430947390630, 9.993621202889750647401436329906

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