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Properties

Label 2-1152-9.7-c1-0-11
Degree $2$
Conductor $1152$
Sign $-0.981 - 0.190i$
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  + (−1.23 + 1.21i)3-s + (2.22 + 3.84i)5-s + (−1.45 + 2.51i)7-s + (0.0524 − 2.99i)9-s + (−1.08 + 1.87i)11-s + (1.96 + 3.40i)13-s + (−7.41 − 2.05i)15-s + 1.79·17-s + 1.76·19-s + (−1.26 − 4.87i)21-s + (3.44 + 5.96i)23-s + (−7.36 + 12.7i)25-s + (3.57 + 3.76i)27-s + (2.87 − 4.97i)29-s + (−3.27 − 5.67i)31-s + ⋯
L(s)  = 1  + (−0.713 + 0.700i)3-s + (0.993 + 1.71i)5-s + (−0.549 + 0.952i)7-s + (0.0174 − 0.999i)9-s + (−0.326 + 0.565i)11-s + (0.544 + 0.943i)13-s + (−1.91 − 0.530i)15-s + 0.435·17-s + 0.405·19-s + (−0.275 − 1.06i)21-s + (0.717 + 1.24i)23-s + (−1.47 + 2.54i)25-s + (0.688 + 0.725i)27-s + (0.533 − 0.924i)29-s + (−0.588 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.981 - 0.190i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.352489187\)
\(L(\frac12)\) \(\approx\) \(1.352489187\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 1.21i)T \)
good5 \( 1 + (-2.22 - 3.84i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.45 - 2.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.08 - 1.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.96 - 3.40i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.79T + 17T^{2} \)
19 \( 1 - 1.76T + 19T^{2} \)
23 \( 1 + (-3.44 - 5.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.87 + 4.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.27 + 5.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.51T + 37T^{2} \)
41 \( 1 + (3.68 + 6.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.53 + 4.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.98 + 8.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 + (2.30 + 3.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.87 - 3.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.36 - 4.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.907T + 71T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 + (1.23 - 2.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.09 - 1.89i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.30T + 89T^{2} \)
97 \( 1 + (-4.45 + 7.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.04534898267126792998475514374, −9.646561597618174114171894533932, −8.974597275134112654513607485553, −7.32232912283646679932609833340, −6.72931919029615318132727420251, −5.80114589736120777094691942304, −5.52517918230243813670022936190, −3.94095510719445290949246448018, −3.02693180186588275793736397595, −2.03227660796377656763210562047, 0.71755508268225049236937556151, 1.29884094125154417974441817333, 2.95405137057805632185617010004, 4.51083908829877049061157206169, 5.26581769347485318961674242877, 5.93097506538591942042454039101, 6.73623697440113847580644464899, 7.87361970574066014718717187145, 8.515772951401929823095088341166, 9.397376027562652629000890459853

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