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Properties

Label 2-165-33.17-c1-0-1
Degree $2$
Conductor $165$
Sign $0.918 + 0.396i$
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  + (−0.796 − 2.45i)2-s + (−0.451 + 1.67i)3-s + (−3.76 + 2.73i)4-s + (0.951 + 0.309i)5-s + (4.46 − 0.226i)6-s + (2.05 + 2.82i)7-s + (5.52 + 4.01i)8-s + (−2.59 − 1.50i)9-s − 2.57i·10-s + (3.28 − 0.490i)11-s + (−2.87 − 7.52i)12-s + (1.33 − 0.433i)13-s + (5.29 − 7.29i)14-s + (−0.945 + 1.45i)15-s + (2.57 − 7.91i)16-s + (−2.35 + 7.24i)17-s + ⋯
L(s)  = 1  + (−0.563 − 1.73i)2-s + (−0.260 + 0.965i)3-s + (−1.88 + 1.36i)4-s + (0.425 + 0.138i)5-s + (1.82 − 0.0924i)6-s + (0.776 + 1.06i)7-s + (1.95 + 1.41i)8-s + (−0.864 − 0.502i)9-s − 0.815i·10-s + (0.989 − 0.147i)11-s + (−0.829 − 2.17i)12-s + (0.370 − 0.120i)13-s + (1.41 − 1.94i)14-s + (−0.244 + 0.374i)15-s + (0.642 − 1.97i)16-s + (−0.570 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.918 + 0.396i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.918 + 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.800734 - 0.165342i\)
\(L(\frac12)\) \(\approx\) \(0.800734 - 0.165342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.451 - 1.67i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-3.28 + 0.490i)T \)
good2 \( 1 + (0.796 + 2.45i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-2.05 - 2.82i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.33 + 0.433i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.35 - 7.24i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.49 + 2.05i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.0193iT - 23T^{2} \)
29 \( 1 + (-1.69 + 1.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.712 + 2.19i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.92 - 4.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.92 + 1.39i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.48iT - 43T^{2} \)
47 \( 1 + (-4.14 + 5.70i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.30 + 1.39i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.94 + 8.17i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.32 - 0.756i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.98T + 67T^{2} \)
71 \( 1 + (5.33 + 1.73i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.35 + 7.37i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.17 - 0.705i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.89 + 5.84i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (4.05 + 12.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.21257115482314198783018464357, −11.56402743751084874682180541103, −10.80558276317137239686864081445, −9.980413229226037294209357597225, −8.858270451912165181330079228504, −8.571332354861390988484237517998, −5.98928990535718089915950966786, −4.58780406404379280600005322542, −3.40604801482175065748987276729, −1.87804929176185695636370844507, 1.13925755068356572498289480774, 4.60868230067172982189892808004, 5.73248475246616882656111091929, 6.93116854526436474312012515878, 7.31901074937192985072723544128, 8.493169513776874045761956535148, 9.373387655299476057348884143110, 10.74459676086905749457418325247, 11.97441813353045747042035541015, 13.57971556376804512513422071444

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