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Properties

Label 2-1152-72.59-c1-0-19
Degree $2$
Conductor $1152$
Sign $0.342 - 0.939i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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 + (−1.41 + 2.44i)5-s + (1.5 − 2.59i)9-s + (4.5 − 2.59i)11-s + (4.24 + 2.44i)13-s − 4.89i·15-s − 1.73i·17-s − 3·19-s + (4.24 − 7.34i)23-s + (−1.49 − 2.59i)25-s + 5.19i·27-s + (2.82 + 4.89i)29-s + (−4.5 + 7.79i)33-s + 4.89i·37-s − 8.48·39-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.632 + 1.09i)5-s + (0.5 − 0.866i)9-s + (1.35 − 0.783i)11-s + (1.17 + 0.679i)13-s − 1.26i·15-s − 0.420i·17-s − 0.688·19-s + (0.884 − 1.53i)23-s + (−0.299 − 0.519i)25-s + 0.999i·27-s + (0.525 + 0.909i)29-s + (−0.783 + 1.35i)33-s + 0.805i·37-s − 1.35·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.193521714\)
\(L(\frac12)\) \(\approx\) \(1.193521714\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.24 - 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-4.24 + 7.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.82 - 4.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.89iT - 37T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.24 - 7.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 + (-4.5 - 2.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.48 - 4.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + (12.7 - 7.34i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9 - 5.19i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.22614870602537645659064223109, −9.040558566265035374506544559950, −8.580402681337371416521878375494, −7.07807048258364687292046484071, −6.57666144807762741697190410026, −6.01594673829508646436149777431, −4.59994272175552682678297190532, −3.87586365739569450190754808460, −3.02953005790363397746601176677, −1.06250292230322306631898930342, 0.801670029554136078794770375776, 1.72440196015286130785042541475, 3.73620157241237739890366136679, 4.45056448115224246455841261352, 5.41702532630897630493696247525, 6.25184919060568999398418839300, 7.09924824556789390416467893181, 8.018754682511173200908277176376, 8.705142543093681600592719891344, 9.586201210496854533877472067866

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