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Properties

Label 2-1152-9.7-c1-0-0
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: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2140290470\)
\(L(\frac12)\) \(\approx\) \(0.2140290470\)
\(L(\frac{3}{2})\) 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

−10.37829353650363508332983178345, −9.557844757515741159230213160543, −8.905901597893839777892052190584, −8.100607362688765996244203833199, −6.62669267223111852132643664133, −6.16035655605890447690196902058, −5.30946515735931912346998922939, −4.30528666899952609294809972785, −2.98998441733124568864269787089, −2.56061080430545217619519186880, 0.088055534671182807993080837585, 1.62453849763483439058780004907, 2.45240118230856383745009262574, 4.36942552278794125216277183542, 4.67702808489542514242146597638, 6.25085085837685530198553658324, 6.76053738934090537645911447906, 7.29135845023970604408750215827, 8.676954226115236751874685477153, 9.025016820480026891969296036868

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