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Properties

Label 2-1152-48.11-c3-0-6
Degree $2$
Conductor $1152$
Sign $-0.462 - 0.886i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.7 − 11.7i)5-s − 11.9·7-s + (36.9 + 36.9i)11-s + (−20.4 + 20.4i)13-s + 81.1i·17-s + (−29.9 − 29.9i)19-s + 163. i·23-s − 151. i·25-s + (−201. − 201. i)29-s − 43.1i·31-s + (−141. + 141. i)35-s + (−100. − 100. i)37-s − 345.·41-s + (−326. + 326. i)43-s + 116.·47-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)5-s − 0.647·7-s + (1.01 + 1.01i)11-s + (−0.436 + 0.436i)13-s + 1.15i·17-s + (−0.361 − 0.361i)19-s + 1.48i·23-s − 1.21i·25-s + (−1.28 − 1.28i)29-s − 0.249i·31-s + (−0.681 + 0.681i)35-s + (−0.444 − 0.444i)37-s − 1.31·41-s + (−1.15 + 1.15i)43-s + 0.361·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.462 - 0.886i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.026965620\)
\(L(\frac12)\) \(\approx\) \(1.026965620\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + (-11.7 + 11.7i)T - 125iT^{2} \)
7 \( 1 + 11.9T + 343T^{2} \)
11 \( 1 + (-36.9 - 36.9i)T + 1.33e3iT^{2} \)
13 \( 1 + (20.4 - 20.4i)T - 2.19e3iT^{2} \)
17 \( 1 - 81.1iT - 4.91e3T^{2} \)
19 \( 1 + (29.9 + 29.9i)T + 6.85e3iT^{2} \)
23 \( 1 - 163. iT - 1.21e4T^{2} \)
29 \( 1 + (201. + 201. i)T + 2.43e4iT^{2} \)
31 \( 1 + 43.1iT - 2.97e4T^{2} \)
37 \( 1 + (100. + 100. i)T + 5.06e4iT^{2} \)
41 \( 1 + 345.T + 6.89e4T^{2} \)
43 \( 1 + (326. - 326. i)T - 7.95e4iT^{2} \)
47 \( 1 - 116.T + 1.03e5T^{2} \)
53 \( 1 + (16.3 - 16.3i)T - 1.48e5iT^{2} \)
59 \( 1 + (-46.4 - 46.4i)T + 2.05e5iT^{2} \)
61 \( 1 + (69.6 - 69.6i)T - 2.26e5iT^{2} \)
67 \( 1 + (157. + 157. i)T + 3.00e5iT^{2} \)
71 \( 1 + 690. iT - 3.57e5T^{2} \)
73 \( 1 - 799. iT - 3.89e5T^{2} \)
79 \( 1 - 763. iT - 4.93e5T^{2} \)
83 \( 1 + (940. - 940. i)T - 5.71e5iT^{2} \)
89 \( 1 - 660.T + 7.04e5T^{2} \)
97 \( 1 + 821.T + 9.12e5T^{2} \)
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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

−9.565082892197055051602082033688, −9.193214166210969458006772560016, −8.180937922498253303591944964452, −7.06527697073704840811575556242, −6.27566999959759821563934677609, −5.50599998104636104306836531544, −4.53620004301803123101064731500, −3.67131219523143009628061392647, −2.03674547345333640331793775205, −1.45861998326910372377348997018, 0.22247987924935603697667502207, 1.77119667834454733506598154781, 2.92468230014974928164529791237, 3.50561911065805113175014827960, 5.02559085512124413335309705232, 5.95448012181534356131097491321, 6.64488124304774279830521404656, 7.15014616264661838749517542743, 8.556453985411779696917959177117, 9.203614178753633218728749117557

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