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Properties

Label 2-165-33.32-c5-0-20
Degree $2$
Conductor $165$
Sign $-0.196 - 0.980i$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.11·2-s + (11.9 + 9.98i)3-s − 30.7·4-s − 25i·5-s + (13.4 + 11.1i)6-s − 234. i·7-s − 70.2·8-s + (43.7 + 239. i)9-s − 27.9i·10-s + (−251. + 312. i)11-s + (−368. − 306. i)12-s + 844. i·13-s − 262. i·14-s + (249. − 299. i)15-s + 905.·16-s + 1.23e3·17-s + ⋯
L(s)  = 1  + 0.197·2-s + (0.768 + 0.640i)3-s − 0.960·4-s − 0.447i·5-s + (0.152 + 0.126i)6-s − 1.80i·7-s − 0.388·8-s + (0.180 + 0.983i)9-s − 0.0885i·10-s + (−0.627 + 0.778i)11-s + (−0.738 − 0.615i)12-s + 1.38i·13-s − 0.357i·14-s + (0.286 − 0.343i)15-s + 0.883·16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ -0.196 - 0.980i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.530311020\)
\(L(\frac12)\) \(\approx\) \(1.530311020\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.9 - 9.98i)T \)
5 \( 1 + 25iT \)
11 \( 1 + (251. - 312. i)T \)
good2 \( 1 - 1.11T + 32T^{2} \)
7 \( 1 + 234. iT - 1.68e4T^{2} \)
13 \( 1 - 844. iT - 3.71e5T^{2} \)
17 \( 1 - 1.23e3T + 1.41e6T^{2} \)
19 \( 1 - 1.29e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.74e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.66e3T + 2.05e7T^{2} \)
31 \( 1 + 8.61e3T + 2.86e7T^{2} \)
37 \( 1 - 180.T + 6.93e7T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
43 \( 1 - 5.26e3iT - 1.47e8T^{2} \)
47 \( 1 - 7.28e3iT - 2.29e8T^{2} \)
53 \( 1 - 5.28e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.77e4iT - 7.14e8T^{2} \)
61 \( 1 + 6.01e3iT - 8.44e8T^{2} \)
67 \( 1 - 1.57e4T + 1.35e9T^{2} \)
71 \( 1 - 5.06e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.19e3iT - 2.07e9T^{2} \)
79 \( 1 - 1.06e5iT - 3.07e9T^{2} \)
83 \( 1 - 7.99e4T + 3.93e9T^{2} \)
89 \( 1 + 9.01e3iT - 5.58e9T^{2} \)
97 \( 1 - 1.20e5T + 8.58e9T^{2} \)
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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

−12.54934369462449423977140940472, −10.96037853092275559636991454584, −9.813283530257772112226103169021, −9.496182550540741342966605108781, −8.050197719518995159164653185546, −7.34366510856249792305225136100, −5.26241488238502748328727157186, −4.23476782038343096006427671985, −3.65292400867787629604332682573, −1.42409195677069942380027702601, 0.45693121560127636356920134560, 2.55704822742216511564519919565, 3.29999567721731828910701248673, 5.28099737659703155328559677245, 6.05578550384844348912588229914, 7.79470693859371737525342196546, 8.559659396443439475094563962310, 9.265100257331586524919751635344, 10.54479270411917682684579838722, 12.10763222778833397067618581688

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