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Properties

Label 2-165-33.17-c1-0-0
Degree $2$
Conductor $165$
Sign $-0.925 + 0.379i$
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  + (0.848 + 2.61i)2-s + (−1.68 + 0.400i)3-s + (−4.48 + 3.25i)4-s + (0.951 + 0.309i)5-s + (−2.47 − 4.06i)6-s + (−0.0372 − 0.0512i)7-s + (−7.86 − 5.71i)8-s + (2.67 − 1.34i)9-s + 2.74i·10-s + (1.43 + 2.98i)11-s + (6.25 − 7.28i)12-s + (−1.24 + 0.402i)13-s + (0.102 − 0.140i)14-s + (−1.72 − 0.139i)15-s + (4.82 − 14.8i)16-s + (−0.915 + 2.81i)17-s + ⋯
L(s)  = 1  + (0.600 + 1.84i)2-s + (−0.972 + 0.231i)3-s + (−2.24 + 1.62i)4-s + (0.425 + 0.138i)5-s + (−1.01 − 1.65i)6-s + (−0.0140 − 0.0193i)7-s + (−2.78 − 2.02i)8-s + (0.893 − 0.449i)9-s + 0.868i·10-s + (0.433 + 0.901i)11-s + (1.80 − 2.10i)12-s + (−0.343 + 0.111i)13-s + (0.0273 − 0.0375i)14-s + (−0.445 − 0.0361i)15-s + (1.20 − 3.71i)16-s + (−0.222 + 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.925 + 0.379i$
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: \((2,\ 165,\ (\ :1/2),\ -0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189664 - 0.961581i\)
\(L(\frac12)\) \(\approx\) \(0.189664 - 0.961581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 0.400i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-1.43 - 2.98i)T \)
good2 \( 1 + (-0.848 - 2.61i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (0.0372 + 0.0512i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.24 - 0.402i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.915 - 2.81i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.152 - 0.209i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.90iT - 23T^{2} \)
29 \( 1 + (-7.27 + 5.28i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.401 - 1.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.16 + 3.02i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.06 - 2.95i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.94iT - 43T^{2} \)
47 \( 1 + (4.62 - 6.37i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.06 + 1.32i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.85 + 2.54i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.07 - 1.32i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (8.12 + 2.64i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.99 + 11.0i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-7.16 + 2.32i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.56 + 4.82i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (-0.276 - 0.850i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.53034678857935849690954377002, −12.68092128512884267003017240674, −11.79732061023598158635341633408, −10.06330410900246202123161700982, −9.209227826000607421178616140473, −7.75651678250588366190095138583, −6.77129840274485484348989660171, −6.03241954934137299965934423719, −4.97927180136183370035734893330, −4.02266498785253238441411910073, 0.950001563269320814917010741719, 2.67215933974128258406166093073, 4.35423111458879285207089271848, 5.31116844849169005823734119943, 6.42064693458455225137976471390, 8.645862266741581043852299532031, 9.775813316126524470290345798683, 10.61027446752595346523023004961, 11.38067163757953251336778312690, 12.17466443676202036585513356703

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