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Properties

Label 2-165-33.8-c1-0-0
Degree $2$
Conductor $165$
Sign $-0.812 + 0.583i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.860 + 0.625i)2-s + (−1.27 + 1.17i)3-s + (−0.268 + 0.825i)4-s + (−0.587 + 0.809i)5-s + (0.365 − 1.80i)6-s + (−2.18 − 0.709i)7-s + (−0.942 − 2.90i)8-s + (0.254 − 2.98i)9-s − 1.06i·10-s + (2.26 + 2.41i)11-s + (−0.625 − 1.36i)12-s + (−1.13 − 1.55i)13-s + (2.32 − 0.754i)14-s + (−0.197 − 1.72i)15-s + (1.22 + 0.887i)16-s + (−3.01 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.608 + 0.442i)2-s + (−0.736 + 0.676i)3-s + (−0.134 + 0.412i)4-s + (−0.262 + 0.361i)5-s + (0.149 − 0.737i)6-s + (−0.825 − 0.268i)7-s + (−0.333 − 1.02i)8-s + (0.0849 − 0.996i)9-s − 0.336i·10-s + (0.683 + 0.729i)11-s + (−0.180 − 0.394i)12-s + (−0.313 − 0.431i)13-s + (0.620 − 0.201i)14-s + (−0.0511 − 0.444i)15-s + (0.305 + 0.221i)16-s + (−0.732 − 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.812 + 0.583i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.812 + 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0670539 - 0.208273i\)
\(L(\frac12)\) \(\approx\) \(0.0670539 - 0.208273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.27 - 1.17i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-2.26 - 2.41i)T \)
good2 \( 1 + (0.860 - 0.625i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (2.18 + 0.709i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.13 + 1.55i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.01 + 2.19i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (8.13 - 2.64i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.81iT - 23T^{2} \)
29 \( 1 + (1.01 - 3.13i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.98 + 2.89i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.59 - 11.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.619 + 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + (3.15 - 1.02i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.55 + 4.88i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0820 - 0.0266i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.79 + 6.60i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.92T + 67T^{2} \)
71 \( 1 + (8.64 - 11.9i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.23 - 1.70i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.0325 - 0.0448i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.22 - 0.886i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.65iT - 89T^{2} \)
97 \( 1 + (-6.60 + 4.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.15338505328010670822749268474, −12.36904298657780706268116355359, −11.37739429565327149982296415272, −10.10579320256522559675484219938, −9.571807124460043026466053301787, −8.370837378874232559656525149989, −6.95313470838101207000940898582, −6.39240316618760759857979794340, −4.52151238210587104915748019722, −3.45640842475743546070719513304, 0.26355480183838096913514197409, 2.15664930172040460877842274761, 4.49126178906887665405393955258, 5.98680440176187495254087186288, 6.68808195995114958009038516345, 8.438112227426208943994736756806, 9.082315097249361157620480135505, 10.45233001376250961754092526727, 11.13027289386697236010932077094, 12.14702728954266766238649887160

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