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Properties

Label 2-165-5.4-c5-0-0
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

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

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