Properties

Degree $2$
Conductor $1152$
Sign $-0.999 + 0.0366i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.24 − 5.24i)5-s − 5.32·7-s + (12.2 − 12.2i)11-s + (5.73 − 5.73i)13-s + 23.3·17-s + (−11.7 − 11.7i)19-s − 5.80·23-s + 29.9i·25-s + (18.3 − 18.3i)29-s + 16.9i·31-s + (27.9 + 27.9i)35-s + (−15.3 − 15.3i)37-s − 29.2i·41-s + (−33.4 + 33.4i)43-s − 18.2i·47-s + ⋯
L(s)  = 1  + (−1.04 − 1.04i)5-s − 0.761·7-s + (1.11 − 1.11i)11-s + (0.441 − 0.441i)13-s + 1.37·17-s + (−0.618 − 0.618i)19-s − 0.252·23-s + 1.19i·25-s + (0.634 − 0.634i)29-s + 0.545i·31-s + (0.798 + 0.798i)35-s + (−0.414 − 0.414i)37-s − 0.713i·41-s + (−0.776 + 0.776i)43-s − 0.387i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.999 + 0.0366i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.999 + 0.0366i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8152212709\)
\(L(\frac12)\) \(\approx\) \(0.8152212709\)
\(L(2)\) 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 \( 1 + (5.24 + 5.24i)T + 25iT^{2} \)
7 \( 1 + 5.32T + 49T^{2} \)
11 \( 1 + (-12.2 + 12.2i)T - 121iT^{2} \)
13 \( 1 + (-5.73 + 5.73i)T - 169iT^{2} \)
17 \( 1 - 23.3T + 289T^{2} \)
19 \( 1 + (11.7 + 11.7i)T + 361iT^{2} \)
23 \( 1 + 5.80T + 529T^{2} \)
29 \( 1 + (-18.3 + 18.3i)T - 841iT^{2} \)
31 \( 1 - 16.9iT - 961T^{2} \)
37 \( 1 + (15.3 + 15.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 29.2iT - 1.68e3T^{2} \)
43 \( 1 + (33.4 - 33.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 18.2iT - 2.20e3T^{2} \)
53 \( 1 + (66.9 + 66.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (27.1 - 27.1i)T - 3.48e3iT^{2} \)
61 \( 1 + (65.2 - 65.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (-37.6 - 37.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 42.6T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 - 21.2iT - 6.24e3T^{2} \)
83 \( 1 + (-24.1 - 24.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 52.8iT - 7.92e3T^{2} \)
97 \( 1 + 21.0T + 9.40e3T^{2} \)
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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

−8.999460728050316247934728681961, −8.461639583214669982874876391865, −7.76007388642676900135394495677, −6.60948814131073847167180270308, −5.85741377717507069083077209709, −4.78883831760456278395257110881, −3.77107184996394961337483830757, −3.20643318524196749929487720024, −1.20755887600626196255508920029, −0.28111922007951131680294158593, 1.57576991536842322614397449043, 3.11957022216601869939887216424, 3.71816721055800771852063441397, 4.60767225403785632956025189254, 6.16424376826008989407363533499, 6.68147272747269653752330808533, 7.45848222967089664831446538096, 8.214413948974070928719868161204, 9.353040820497059997142914540850, 10.00588668350089354710293209858

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