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Properties

Label 2-24e2-9.4-c1-0-19
Degree $2$
Conductor $576$
Sign $0.173 + 0.984i$
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  + (1.5 − 0.866i)3-s + (2 − 3.46i)5-s + (−1 − 1.73i)7-s + (1.5 − 2.59i)9-s + (2.5 + 4.33i)11-s + (−1 + 1.73i)13-s − 6.92i·15-s − 3·17-s + 19-s + (−3 − 1.73i)21-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s − 5.19i·27-s + (−1 − 1.73i)29-s + (−2 + 3.46i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.894 − 1.54i)5-s + (−0.377 − 0.654i)7-s + (0.5 − 0.866i)9-s + (0.753 + 1.30i)11-s + (−0.277 + 0.480i)13-s − 1.78i·15-s − 0.727·17-s + 0.229·19-s + (−0.654 − 0.377i)21-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s − 0.999i·27-s + (−0.185 − 0.321i)29-s + (−0.359 + 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66105 - 1.39379i\)
\(L(\frac12)\) \(\approx\) \(1.66105 - 1.39379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.03146618480358117410861531935, −9.442396673457230160521912035213, −9.048238395078110363851809613977, −7.895419256837656918671710299719, −7.03043548563874145731304219600, −6.07519288500321125832662026857, −4.69911344198151850806815910000, −3.97215094517217592544015534420, −2.16571508983059019560353642458, −1.27072654520494929129586247714, 2.29476337896407485069167871132, 2.94200606822705141090101637577, 3.95124916271460785385898270915, 5.64174789599990200284470977679, 6.32297896590250960601168403954, 7.30165458041020094032100945604, 8.490267108338694143551235644450, 9.243217227527519171455507767977, 10.00726799784679649735215587144, 10.75517737425482418044524040959

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