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Properties

Label 2-1152-48.35-c3-0-12
Degree $2$
Conductor $1152$
Sign $-0.962 - 0.272i$
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  + (11.2 + 11.2i)5-s − 19.1·7-s + (−34.3 + 34.3i)11-s + (59.4 + 59.4i)13-s + 102. i·17-s + (60.4 − 60.4i)19-s − 106. i·23-s + 130. i·25-s + (−30.6 + 30.6i)29-s + 99.7i·31-s + (−216. − 216. i)35-s + (94.9 − 94.9i)37-s + 34.5·41-s + (198. + 198. i)43-s − 314.·47-s + ⋯
L(s)  = 1  + (1.01 + 1.01i)5-s − 1.03·7-s + (−0.941 + 0.941i)11-s + (1.26 + 1.26i)13-s + 1.46i·17-s + (0.729 − 0.729i)19-s − 0.961i·23-s + 1.04i·25-s + (−0.196 + 0.196i)29-s + 0.578i·31-s + (−1.04 − 1.04i)35-s + (0.421 − 0.421i)37-s + 0.131·41-s + (0.702 + 0.702i)43-s − 0.977·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.962 - 0.272i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.962 - 0.272i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.498353131\)
\(L(\frac12)\) \(\approx\) \(1.498353131\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.2 - 11.2i)T + 125iT^{2} \)
7 \( 1 + 19.1T + 343T^{2} \)
11 \( 1 + (34.3 - 34.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (-59.4 - 59.4i)T + 2.19e3iT^{2} \)
17 \( 1 - 102. iT - 4.91e3T^{2} \)
19 \( 1 + (-60.4 + 60.4i)T - 6.85e3iT^{2} \)
23 \( 1 + 106. iT - 1.21e4T^{2} \)
29 \( 1 + (30.6 - 30.6i)T - 2.43e4iT^{2} \)
31 \( 1 - 99.7iT - 2.97e4T^{2} \)
37 \( 1 + (-94.9 + 94.9i)T - 5.06e4iT^{2} \)
41 \( 1 - 34.5T + 6.89e4T^{2} \)
43 \( 1 + (-198. - 198. i)T + 7.95e4iT^{2} \)
47 \( 1 + 314.T + 1.03e5T^{2} \)
53 \( 1 + (187. + 187. i)T + 1.48e5iT^{2} \)
59 \( 1 + (382. - 382. i)T - 2.05e5iT^{2} \)
61 \( 1 + (509. + 509. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-107. + 107. i)T - 3.00e5iT^{2} \)
71 \( 1 - 209. iT - 3.57e5T^{2} \)
73 \( 1 - 411. iT - 3.89e5T^{2} \)
79 \( 1 + 992. iT - 4.93e5T^{2} \)
83 \( 1 + (-852. - 852. i)T + 5.71e5iT^{2} \)
89 \( 1 + 205.T + 7.04e5T^{2} \)
97 \( 1 + 775.T + 9.12e5T^{2} \)
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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

−9.724532360997946315893518265375, −9.278790375660019815320631748351, −8.166379222863102137334966717802, −7.00355219760490784719721064488, −6.45404671140725722981172936847, −5.90240242471382806648446489515, −4.60655027338982876034519183189, −3.46599570979141230567110185318, −2.55121652356723143891411906529, −1.60229412027580748114034302498, 0.35828934464862104755171606166, 1.25993217141761391093863753417, 2.81491546031911662275078985399, 3.49140758016674258006317588844, 5.04320940048754261568212302913, 5.72330815924436131365025661724, 6.13175567855415745706483861245, 7.56456141162263006426033870607, 8.272407725397594910269388141039, 9.257952840371232752703686574350

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