Properties

Label 2-1152-72.59-c1-0-15
Degree $2$
Conductor $1152$
Sign $0.191 - 0.981i$
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  + (−0.944 + 1.45i)3-s + (0.637 − 1.10i)5-s + (−1.73 + 1.00i)7-s + (−1.21 − 2.74i)9-s + (0.511 − 0.295i)11-s + (2.70 + 1.56i)13-s + (1.00 + 1.96i)15-s − 4.34i·17-s + 4.36·19-s + (0.183 − 3.46i)21-s + (−4.02 + 6.97i)23-s + (1.68 + 2.92i)25-s + (5.13 + 0.821i)27-s + (3.39 + 5.87i)29-s + (0.688 + 0.397i)31-s + ⋯
L(s)  = 1  + (−0.545 + 0.838i)3-s + (0.285 − 0.493i)5-s + (−0.655 + 0.378i)7-s + (−0.405 − 0.913i)9-s + (0.154 − 0.0891i)11-s + (0.750 + 0.433i)13-s + (0.258 + 0.508i)15-s − 1.05i·17-s + 1.00·19-s + (0.0400 − 0.756i)21-s + (−0.839 + 1.45i)23-s + (0.337 + 0.584i)25-s + (0.987 + 0.158i)27-s + (0.629 + 1.09i)29-s + (0.123 + 0.0713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.191 - 0.981i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.191 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.222071188\)
\(L(\frac12)\) \(\approx\) \(1.222071188\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.944 - 1.45i)T \)
good5 \( 1 + (-0.637 + 1.10i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.73 - 1.00i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.511 + 0.295i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.70 - 1.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.34iT - 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 + (4.02 - 6.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.39 - 5.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.688 - 0.397i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.791iT - 37T^{2} \)
41 \( 1 + (3.81 + 2.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 - 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.881 + 1.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.01T + 53T^{2} \)
59 \( 1 + (-6.66 - 3.84i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.9 - 6.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.968 + 1.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.859T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + (14.6 - 8.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.89 + 3.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.00iT - 89T^{2} \)
97 \( 1 + (-5.03 - 8.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.797995156704488251852329837154, −9.277151477310030608061272896087, −8.745835148476646632545025962830, −7.40017597673940726496706570960, −6.43611400284537014814430672434, −5.59019043783178333595571558493, −5.01271283271743526575932618281, −3.82627181348214562909991806739, −3.01007888875039467282730361579, −1.18614398747458404343329844008, 0.66905859295265219488452357855, 2.10111276145554378483950766235, 3.22848986574827471115595979640, 4.40867978631090214688825122064, 5.72597137859096158967758629099, 6.34287912757831714864868984831, 6.87775552501180461451446447015, 7.966999870484995321163136572614, 8.545896234319728839769294986015, 9.945689901370590791638596222969

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