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Properties

Label 2-165-11.3-c1-0-4
Degree $2$
Conductor $165$
Sign $0.999 - 0.0335i$
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  + (1.12 − 0.817i)2-s + (0.309 + 0.951i)3-s + (−0.0207 + 0.0638i)4-s + (0.809 + 0.587i)5-s + (1.12 + 0.817i)6-s + (0.394 − 1.21i)7-s + (0.888 + 2.73i)8-s + (−0.809 + 0.587i)9-s + 1.39·10-s + (−1.20 − 3.09i)11-s − 0.0671·12-s + (1.14 − 0.833i)13-s + (−0.548 − 1.68i)14-s + (−0.309 + 0.951i)15-s + (3.12 + 2.26i)16-s + (−4.04 − 2.93i)17-s + ⋯
L(s)  = 1  + (0.795 − 0.577i)2-s + (0.178 + 0.549i)3-s + (−0.0103 + 0.0319i)4-s + (0.361 + 0.262i)5-s + (0.459 + 0.333i)6-s + (0.149 − 0.459i)7-s + (0.313 + 0.966i)8-s + (−0.269 + 0.195i)9-s + 0.439·10-s + (−0.362 − 0.931i)11-s − 0.0193·12-s + (0.318 − 0.231i)13-s + (−0.146 − 0.451i)14-s + (−0.0797 + 0.245i)15-s + (0.780 + 0.567i)16-s + (−0.981 − 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0335i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.999 - 0.0335i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74780 + 0.0292900i\)
\(L(\frac12)\) \(\approx\) \(1.74780 + 0.0292900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (1.20 + 3.09i)T \)
good2 \( 1 + (-1.12 + 0.817i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.394 + 1.21i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.14 + 0.833i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.04 + 2.93i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.0488 - 0.150i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.00T + 23T^{2} \)
29 \( 1 + (-1.93 + 5.96i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.46 - 1.79i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.45 - 4.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.34 - 7.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.41T + 43T^{2} \)
47 \( 1 + (2.54 + 7.82i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.57 - 5.50i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.50 + 7.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.5 - 8.40i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.38T + 67T^{2} \)
71 \( 1 + (-5.48 - 3.98i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.67 + 8.23i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.05 + 1.49i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.18 + 5.94i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (5.18 - 3.76i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.08733662520105790859482675472, −11.71222861497629655481165089172, −11.01984542765538777772291473764, −10.12314177870620590806962710868, −8.776312767770115974750943690072, −7.78567826921000256461213435481, −6.09712739101294474245163991522, −4.85204285090631911663776635874, −3.73937368565990742729851622335, −2.55157414298016030211126324056, 1.98713927014760650698380064284, 4.11416317660694596950011163014, 5.33824712939217707940080857818, 6.30479391735833090696095724856, 7.30545035099355614585624763607, 8.627603981569154425380361406084, 9.697362969871592179955794689032, 10.88616746591835818724900510433, 12.41882239487876812004888077604, 12.85284299778083186176505701978

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