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Properties

Label 2-165-33.32-c5-0-22
Degree $2$
Conductor $165$
Sign $-0.729 - 0.683i$
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  − 7.77·2-s + (−12.8 − 8.88i)3-s + 28.3·4-s + 25i·5-s + (99.5 + 69.0i)6-s + 223. i·7-s + 28.1·8-s + (85.2 + 227. i)9-s − 194. i·10-s + (−58.7 + 396. i)11-s + (−363. − 252. i)12-s + 123. i·13-s − 1.73e3i·14-s + (222. − 320. i)15-s − 1.12e3·16-s + 1.79e3·17-s + ⋯
L(s)  = 1  − 1.37·2-s + (−0.821 − 0.569i)3-s + 0.886·4-s + 0.447i·5-s + (1.12 + 0.782i)6-s + 1.72i·7-s + 0.155·8-s + (0.350 + 0.936i)9-s − 0.614i·10-s + (−0.146 + 0.989i)11-s + (−0.728 − 0.505i)12-s + 0.202i·13-s − 2.36i·14-s + (0.254 − 0.367i)15-s − 1.10·16-s + 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.729 - 0.683i$
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: \((2,\ 165,\ (\ :5/2),\ -0.729 - 0.683i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5899079556\)
\(L(\frac12)\) \(\approx\) \(0.5899079556\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.8 + 8.88i)T \)
5 \( 1 - 25iT \)
11 \( 1 + (58.7 - 396. i)T \)
good2 \( 1 + 7.77T + 32T^{2} \)
7 \( 1 - 223. iT - 1.68e4T^{2} \)
13 \( 1 - 123. iT - 3.71e5T^{2} \)
17 \( 1 - 1.79e3T + 1.41e6T^{2} \)
19 \( 1 + 347. iT - 2.47e6T^{2} \)
23 \( 1 - 3.28e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.29e3T + 2.05e7T^{2} \)
31 \( 1 + 5.94e3T + 2.86e7T^{2} \)
37 \( 1 - 9.52e3T + 6.93e7T^{2} \)
41 \( 1 + 1.89e3T + 1.15e8T^{2} \)
43 \( 1 - 1.17e4iT - 1.47e8T^{2} \)
47 \( 1 - 8.17e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.57e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.85e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.64e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.93e4T + 1.35e9T^{2} \)
71 \( 1 - 3.38e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.12e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.28e4iT - 3.07e9T^{2} \)
83 \( 1 - 9.23e4T + 3.93e9T^{2} \)
89 \( 1 + 1.07e5iT - 5.58e9T^{2} \)
97 \( 1 - 5.28e4T + 8.58e9T^{2} \)
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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

−12.00724984389758415819995394349, −11.29661679950480166243389873063, −10.09126328895359047291787428922, −9.390034844927991158300917494787, −8.124971452860489183995997724092, −7.33285021165011956218843700558, −6.14316457408763511977648483613, −5.03500750569625219084108776540, −2.48942209627586383835269896049, −1.32948499463833568652390227279, 0.49987361665686612930082283186, 0.964227893312152261784223203385, 3.70296047777453028900623473409, 4.92269807720371356148436411693, 6.43054561081531145548650843233, 7.58943786896512365691662995500, 8.517176288305647013846641040648, 9.788645476610142060614282284998, 10.41883058775807895189374483882, 10.97469893731199716412965030620

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