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Properties

Label 2-1152-9.7-c1-0-37
Degree $2$
Conductor $1152$
Sign $-0.684 + 0.729i$
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  + (0.494 − 1.66i)3-s + (0.268 + 0.464i)5-s + (−2.35 + 4.07i)7-s + (−2.51 − 1.64i)9-s + (2.59 − 4.50i)11-s + (−0.778 − 1.34i)13-s + (0.903 − 0.215i)15-s + 0.695·17-s − 5.80·19-s + (5.59 + 5.91i)21-s + (−4.42 − 7.66i)23-s + (2.35 − 4.08i)25-s + (−3.96 + 3.35i)27-s + (1.92 − 3.32i)29-s + (−2.77 − 4.79i)31-s + ⋯
L(s)  = 1  + (0.285 − 0.958i)3-s + (0.119 + 0.207i)5-s + (−0.888 + 1.53i)7-s + (−0.837 − 0.546i)9-s + (0.783 − 1.35i)11-s + (−0.215 − 0.373i)13-s + (0.233 − 0.0556i)15-s + 0.168·17-s − 1.33·19-s + (1.22 + 1.29i)21-s + (−0.923 − 1.59i)23-s + (0.471 − 0.816i)25-s + (−0.763 + 0.646i)27-s + (0.356 − 0.618i)29-s + (−0.497 − 0.861i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.684 + 0.729i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.062358864\)
\(L(\frac12)\) \(\approx\) \(1.062358864\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.494 + 1.66i)T \)
good5 \( 1 + (-0.268 - 0.464i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.35 - 4.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 4.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.778 + 1.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.695T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 + (4.42 + 7.66i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.92 + 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.77 + 4.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + (-1.01 - 1.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.71 + 6.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.186 + 0.322i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.30T + 53T^{2} \)
59 \( 1 + (-2.57 - 4.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.921 - 1.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.79 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 4.40T + 73T^{2} \)
79 \( 1 + (3.32 - 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.28 - 9.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.30T + 89T^{2} \)
97 \( 1 + (7.81 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.195861861762444084815683930267, −8.611077331031578407459862321749, −8.119526327130378496242281638857, −6.71503076926841016454827378653, −6.13518426005918537319594211077, −5.77183824249724211611177054898, −4.02873987493840245009272607072, −2.79517755234655602849172666422, −2.30739428048257510362625840063, −0.42807113592232723154266260022, 1.64727504696856788219278626238, 3.22475963401527668330255589047, 4.11941802224299750455628464065, 4.54949680737722916937236758756, 5.85714877635751600247755189646, 6.95634400760236453194819461720, 7.46064845292419163297666774667, 8.714714002852858002282363792621, 9.550273667693904620682449452654, 9.925493232493281450658527485025

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