Properties

Label 2-165-5.4-c5-0-49
Degree $2$
Conductor $165$
Sign $0.398 - 0.917i$
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  − 8.57i·2-s + 9i·3-s − 41.5·4-s + (−51.2 − 22.2i)5-s + 77.1·6-s − 178. i·7-s + 81.7i·8-s − 81·9-s + (−191. + 439. i)10-s + 121·11-s − 373. i·12-s − 361. i·13-s − 1.53e3·14-s + (200. − 461. i)15-s − 627.·16-s − 934. i·17-s + ⋯
L(s)  = 1  − 1.51i·2-s + 0.577i·3-s − 1.29·4-s + (−0.917 − 0.398i)5-s + 0.875·6-s − 1.37i·7-s + 0.451i·8-s − 0.333·9-s + (−0.604 + 1.39i)10-s + 0.301·11-s − 0.749i·12-s − 0.594i·13-s − 2.08·14-s + (0.230 − 0.529i)15-s − 0.613·16-s − 0.783i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.398 - 0.917i$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ 0.398 - 0.917i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3200166294\)
\(L(\frac12)\) \(\approx\) \(0.3200166294\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9iT \)
5 \( 1 + (51.2 + 22.2i)T \)
11 \( 1 - 121T \)
good2 \( 1 + 8.57iT - 32T^{2} \)
7 \( 1 + 178. iT - 1.68e4T^{2} \)
13 \( 1 + 361. iT - 3.71e5T^{2} \)
17 \( 1 + 934. iT - 1.41e6T^{2} \)
19 \( 1 + 753.T + 2.47e6T^{2} \)
23 \( 1 - 3.23e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.60e3T + 2.05e7T^{2} \)
31 \( 1 - 662.T + 2.86e7T^{2} \)
37 \( 1 - 1.29e4iT - 6.93e7T^{2} \)
41 \( 1 + 2.53e3T + 1.15e8T^{2} \)
43 \( 1 - 2.20e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.08e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.74e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.75e3T + 7.14e8T^{2} \)
61 \( 1 - 3.84e4T + 8.44e8T^{2} \)
67 \( 1 + 3.55e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.26e4T + 1.80e9T^{2} \)
73 \( 1 - 6.89e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.73e4T + 3.07e9T^{2} \)
83 \( 1 + 8.90e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.29e5T + 5.58e9T^{2} \)
97 \( 1 - 1.36e5iT - 8.58e9T^{2} \)
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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

−11.25785379318608577988074749187, −10.25955987307586765203065064049, −9.538374652198089763940682947137, −8.255775895858705937681460112483, −7.05379391909346640092963072474, −4.90182396326538899279501581705, −3.96568734328115536601932277845, −3.20594100869549149885628519183, −1.21391557670851474598101385131, −0.11509938716634567621909878965, 2.34576037626535473995069637567, 4.22598747776920960041577681224, 5.68464686993745955201972787724, 6.50362618916263413397023855597, 7.41178249281050870691671826606, 8.465827466047298349766924390833, 8.973031619092679648871348090195, 10.90365004945107345565947337955, 11.98085564244909646861514371667, 12.71104178507747475764860129749

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