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Properties

Label 2-1152-9.4-c1-0-33
Degree $2$
Conductor $1152$
Sign $0.565 + 0.824i$
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  + (0.366 + 1.69i)3-s + (−1.05 + 1.82i)5-s + (−1.43 − 2.49i)7-s + (−2.73 + 1.24i)9-s + (−1.21 − 2.10i)11-s + (3.30 − 5.71i)13-s + (−3.47 − 1.11i)15-s − 7.56·17-s + 6.25·19-s + (3.69 − 3.35i)21-s + (2.63 − 4.56i)23-s + (0.275 + 0.476i)25-s + (−3.10 − 4.16i)27-s + (−1.57 − 2.73i)29-s + (1.79 − 3.10i)31-s + ⋯
L(s)  = 1  + (0.211 + 0.977i)3-s + (−0.471 + 0.816i)5-s + (−0.543 − 0.942i)7-s + (−0.910 + 0.414i)9-s + (−0.365 − 0.633i)11-s + (0.915 − 1.58i)13-s + (−0.898 − 0.287i)15-s − 1.83·17-s + 1.43·19-s + (0.805 − 0.731i)21-s + (0.549 − 0.952i)23-s + (0.0550 + 0.0953i)25-s + (−0.597 − 0.801i)27-s + (−0.293 − 0.507i)29-s + (0.321 − 0.556i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.565 + 0.824i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9290831267\)
\(L(\frac12)\) \(\approx\) \(0.9290831267\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.366 - 1.69i)T \)
good5 \( 1 + (1.05 - 1.82i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.43 + 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.21 + 2.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.30 + 5.71i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.56T + 17T^{2} \)
19 \( 1 - 6.25T + 19T^{2} \)
23 \( 1 + (-2.63 + 4.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.57 + 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.70T + 37T^{2} \)
41 \( 1 + (1.74 - 3.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.12 - 5.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.32 + 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.953T + 53T^{2} \)
59 \( 1 + (-4.84 + 8.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.57 - 4.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.949 + 1.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.82T + 71T^{2} \)
73 \( 1 + 5.01T + 73T^{2} \)
79 \( 1 + (6.49 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.54 - 2.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.95T + 89T^{2} \)
97 \( 1 + (5.51 + 9.54i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.886747743365929316397240200837, −8.834972525389482150302926443923, −8.120323453701455497726340050434, −7.23356131368640940101317242480, −6.31720462888106258711433614346, −5.34555621036054275732183164464, −4.23926718644717028373116612049, −3.35058891386969276246591932370, −2.86963019502230193840102647105, −0.40315263537687332736287127391, 1.41619790533309068740610335129, 2.44082929141205425080283484235, 3.68304941979844437329025640987, 4.84349182812139540182037831371, 5.77034160115929716504646122497, 6.80967164773980683357986923629, 7.25813238516557211280436593330, 8.623257405866127101476167721035, 8.826398533493579999191482363145, 9.508667774622824450066890336589

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