Properties

Label 2-24e2-9.7-c1-0-19
Degree $2$
Conductor $576$
Sign $-0.594 + 0.804i$
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  + (1.18 − 1.26i)3-s + (−1.68 − 2.92i)5-s + (0.686 − 1.18i)7-s + (−0.186 − 2.99i)9-s + (0.5 − 0.866i)11-s + (2.68 + 4.65i)13-s + (−5.68 − 1.33i)15-s + 0.372·17-s − 6.37·19-s + (−0.686 − 2.27i)21-s + (−2.68 − 4.65i)23-s + (−3.18 + 5.51i)25-s + (−4.00 − 3.31i)27-s + (0.686 − 1.18i)29-s + (−0.313 − 0.543i)31-s + ⋯
L(s)  = 1  + (0.684 − 0.728i)3-s + (−0.754 − 1.30i)5-s + (0.259 − 0.449i)7-s + (−0.0620 − 0.998i)9-s + (0.150 − 0.261i)11-s + (0.745 + 1.29i)13-s + (−1.46 − 0.344i)15-s + 0.0902·17-s − 1.46·19-s + (−0.149 − 0.496i)21-s + (−0.560 − 0.970i)23-s + (−0.637 + 1.10i)25-s + (−0.769 − 0.638i)27-s + (0.127 − 0.220i)29-s + (−0.0563 − 0.0976i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.594 + 0.804i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678353 - 1.34415i\)
\(L(\frac12)\) \(\approx\) \(0.678353 - 1.34415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 1.26i)T \)
good5 \( 1 + (1.68 + 2.92i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.686 + 1.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.68 - 4.65i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 + (2.68 + 4.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.686 + 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.313 + 0.543i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.74T + 37T^{2} \)
41 \( 1 + (-0.127 - 0.221i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.87 + 8.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.686 + 1.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.87 - 6.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + (-0.313 + 0.543i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.68 + 13.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-4.87 + 8.43i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.44072459843414715444952837624, −8.980129765778150270013430790979, −8.727763246085499249556116870882, −7.941038416774272864125033874579, −6.95574285787064920799539087091, −5.98737232361202366826426560155, −4.34377646760899051440474906343, −3.96687817422801291273168861651, −2.10789157058983036020927388752, −0.810704621221195081761531485835, 2.34716446829473048557328915308, 3.35449600559463227719451255998, 4.11006399251522869937958301214, 5.46898259604252434834698099105, 6.58288246111339101907755245212, 7.78779782808171112512966871612, 8.209464148200530820515127692093, 9.300212503901636284253147105014, 10.37538350504438032934305369069, 10.81339833182346947573777189244

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