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Properties

Label 2-1152-9.4-c1-0-39
Degree $2$
Conductor $1152$
Sign $-0.470 + 0.882i$
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  + (−0.564 + 1.63i)3-s + (1.59 − 2.75i)5-s + (−0.607 − 1.05i)7-s + (−2.36 − 1.84i)9-s + (−0.312 − 0.540i)11-s + (1.06 − 1.84i)13-s + (3.61 + 4.16i)15-s − 1.83·17-s − 7.15·19-s + (2.06 − 0.400i)21-s + (−0.780 + 1.35i)23-s + (−2.57 − 4.46i)25-s + (4.36 − 2.82i)27-s + (−4.87 − 8.44i)29-s + (−3.32 + 5.75i)31-s + ⋯
L(s)  = 1  + (−0.325 + 0.945i)3-s + (0.712 − 1.23i)5-s + (−0.229 − 0.397i)7-s + (−0.787 − 0.616i)9-s + (−0.0941 − 0.163i)11-s + (0.295 − 0.511i)13-s + (0.934 + 1.07i)15-s − 0.445·17-s − 1.64·19-s + (0.450 − 0.0874i)21-s + (−0.162 + 0.282i)23-s + (−0.515 − 0.892i)25-s + (0.839 − 0.543i)27-s + (−0.905 − 1.56i)29-s + (−0.596 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7848044957\)
\(L(\frac12)\) \(\approx\) \(0.7848044957\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.564 - 1.63i)T \)
good5 \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.607 + 1.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.312 + 0.540i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.06 + 1.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 + 7.15T + 19T^{2} \)
23 \( 1 + (0.780 - 1.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.87 + 8.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.32 - 5.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.73T + 37T^{2} \)
41 \( 1 + (5.64 - 9.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.51 + 7.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.36 - 2.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.60T + 53T^{2} \)
59 \( 1 + (-4.02 + 6.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.79 - 4.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.95 + 6.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + 5.66T + 73T^{2} \)
79 \( 1 + (3.21 + 5.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.27 - 5.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.02T + 89T^{2} \)
97 \( 1 + (4.70 + 8.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.630347063431837308870237879254, −8.695903733410285975545961333312, −8.367135006332332509329587074601, −6.81865420115524344737063955844, −5.89006794565189735249005019692, −5.23833706720537758326660596530, −4.38522292353542842206147099054, −3.53434928752844361167789495180, −1.95795350436743480965429741904, −0.32609309655743449940316459350, 1.90339160446929825628561439847, 2.46281644344954194888477835391, 3.75621848230263906289071294349, 5.27111189107192057911236418215, 6.10587271385869591996195088158, 6.73743093233582708763113383290, 7.23028969072328662231621490694, 8.472755166914601620803566255810, 9.122368671487082071562155753385, 10.35309272270995043457990928907

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