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Properties

Label 2-165-11.10-c2-0-2
Degree $2$
Conductor $165$
Sign $-0.944 - 0.327i$
Analytic cond. $4.49592$
Root an. cond. $2.12035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.33i·2-s − 1.73·3-s − 1.43·4-s + 2.23·5-s − 4.03i·6-s + 6.32i·7-s + 5.98i·8-s + 2.99·9-s + 5.21i·10-s + (−10.3 − 3.59i)11-s + 2.48·12-s + 17.3i·13-s − 14.7·14-s − 3.87·15-s − 19.6·16-s − 12.2i·17-s + ⋯
L(s)  = 1  + 1.16i·2-s − 0.577·3-s − 0.358·4-s + 0.447·5-s − 0.672i·6-s + 0.903i·7-s + 0.747i·8-s + 0.333·9-s + 0.521i·10-s + (−0.944 − 0.327i)11-s + 0.206·12-s + 1.33i·13-s − 1.05·14-s − 0.258·15-s − 1.22·16-s − 0.721i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(4.49592\)
Root analytic conductor: \(2.12035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1),\ -0.944 - 0.327i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.196682 + 1.16888i\)
\(L(\frac12)\) \(\approx\) \(0.196682 + 1.16888i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
5 \( 1 - 2.23T \)
11 \( 1 + (10.3 + 3.59i)T \)
good2 \( 1 - 2.33iT - 4T^{2} \)
7 \( 1 - 6.32iT - 49T^{2} \)
13 \( 1 - 17.3iT - 169T^{2} \)
17 \( 1 + 12.2iT - 289T^{2} \)
19 \( 1 - 10.6iT - 361T^{2} \)
23 \( 1 + 9.16T + 529T^{2} \)
29 \( 1 - 0.592iT - 841T^{2} \)
31 \( 1 - 17.1T + 961T^{2} \)
37 \( 1 - 55.3T + 1.36e3T^{2} \)
41 \( 1 - 13.5iT - 1.68e3T^{2} \)
43 \( 1 - 36.0iT - 1.84e3T^{2} \)
47 \( 1 + 19.2T + 2.20e3T^{2} \)
53 \( 1 - 70.4T + 2.80e3T^{2} \)
59 \( 1 - 14.9T + 3.48e3T^{2} \)
61 \( 1 + 36.1iT - 3.72e3T^{2} \)
67 \( 1 - 127.T + 4.48e3T^{2} \)
71 \( 1 + 27.8T + 5.04e3T^{2} \)
73 \( 1 - 61.3iT - 5.32e3T^{2} \)
79 \( 1 + 141. iT - 6.24e3T^{2} \)
83 \( 1 + 137. iT - 6.88e3T^{2} \)
89 \( 1 - 160.T + 7.92e3T^{2} \)
97 \( 1 + 57.7T + 9.40e3T^{2} \)
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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

−13.22971840120075364148497297598, −11.93663572630395409382244553199, −11.22879405144248245082392023788, −9.843462875364832647370166440955, −8.759875334595979662179591463262, −7.67500889092948592097917613379, −6.48641772581193411127956978622, −5.75889233289628719401254351705, −4.77984216766363783416506044750, −2.35009472004894073417362134891, 0.78130064227654470655000516000, 2.52205369878817402016781901319, 4.01927467873555351213078770744, 5.43428320436914930603094607267, 6.79422010238449767392057279070, 7.981375538258392498976208975350, 9.722731826990837190766097740416, 10.45978126756726360547285116685, 10.88023197977294239896148875944, 12.12470459249921191699312430453

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