Properties

Label 2-1152-9.7-c1-0-9
Degree $2$
Conductor $1152$
Sign $-0.758 - 0.651i$
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  + (0.857 + 1.50i)3-s + (0.551 + 0.955i)5-s + (−1.62 + 2.81i)7-s + (−1.53 + 2.58i)9-s + (1.28 − 2.23i)11-s + (1.58 + 2.74i)13-s + (−0.965 + 1.64i)15-s + 4.71·17-s − 5.75·19-s + (−5.62 − 0.0333i)21-s + (2.35 + 4.07i)23-s + (1.89 − 3.27i)25-s + (−5.19 − 0.0922i)27-s + (−3.66 + 6.34i)29-s + (−2.93 − 5.07i)31-s + ⋯
L(s)  = 1  + (0.494 + 0.868i)3-s + (0.246 + 0.427i)5-s + (−0.614 + 1.06i)7-s + (−0.510 + 0.860i)9-s + (0.388 − 0.672i)11-s + (0.440 + 0.762i)13-s + (−0.249 + 0.425i)15-s + 1.14·17-s − 1.32·19-s + (−1.22 − 0.00726i)21-s + (0.490 + 0.850i)23-s + (0.378 − 0.655i)25-s + (−0.999 − 0.0177i)27-s + (−0.680 + 1.17i)29-s + (−0.526 − 0.911i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.758 - 0.651i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.661349434\)
\(L(\frac12)\) \(\approx\) \(1.661349434\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.857 - 1.50i)T \)
good5 \( 1 + (-0.551 - 0.955i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.62 - 2.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 + 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 2.74i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.71T + 17T^{2} \)
19 \( 1 + 5.75T + 19T^{2} \)
23 \( 1 + (-2.35 - 4.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.66 - 6.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.93 + 5.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.0714T + 37T^{2} \)
41 \( 1 + (1.63 + 2.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.12 - 3.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (4.19 + 7.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.66 + 8.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.09 - 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.335T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + (-4.85 + 8.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.07 - 5.31i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.42T + 89T^{2} \)
97 \( 1 + (-6.39 + 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.902949165447888261105281275386, −9.283767347816048528622648594068, −8.739914721303071099431650687848, −7.87886582308166341495812567363, −6.56205348459155098530629439815, −5.92191481061467514822745042479, −5.00621237302246166215287861923, −3.73542687399400368798887209635, −3.10603641743979422039113615870, −1.98501066536959533752920786911, 0.67448084148700229230792840174, 1.81948569632725538247638477868, 3.19851835063165924373975855673, 4.00045562624906293218411904112, 5.29065709220841669486880828696, 6.39009917014403236487116484259, 6.96876323631227099894886096117, 7.82454592490932933622416432163, 8.564168009095409984651099092757, 9.441615753459511643243434579347

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