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Properties

Label 2-165-33.17-c1-0-15
Degree $2$
Conductor $165$
Sign $-0.234 - 0.972i$
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 + (−0.139 − 1.72i)3-s + (−4.48 + 3.25i)4-s + (−0.951 − 0.309i)5-s + (−4.39 + 1.83i)6-s + (−0.0372 − 0.0512i)7-s + (7.86 + 5.71i)8-s + (−2.96 + 0.483i)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 + (−0.400 + 1.68i)15-s + (4.82 − 14.8i)16-s + (0.915 − 2.81i)17-s + ⋯
L(s)  = 1  + (−0.600 − 1.84i)2-s + (−0.0807 − 0.996i)3-s + (−2.24 + 1.62i)4-s + (−0.425 − 0.138i)5-s + (−1.79 + 0.747i)6-s + (−0.0140 − 0.0193i)7-s + (2.78 + 2.02i)8-s + (−0.986 + 0.161i)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 + (−0.103 + 0.435i)15-s + (1.20 − 3.71i)16-s + (0.222 − 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.234 - 0.972i$
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: \((2,\ 165,\ (\ :1/2),\ -0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.279449 + 0.354997i\)
\(L(\frac12)\) \(\approx\) \(0.279449 + 0.354997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.139 + 1.72i)T \)
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

−12.01716628416856629862849229794, −11.24901417975305186789392790545, −10.44802274973432936405969126682, −9.053442060174043754866571021557, −8.340053343088852120139803324094, −7.29889978406236163301868136139, −5.20182851478484780688029479090, −3.51907365605501114846932400584, −2.28867962919115219186564045633, −0.52965401122267329829826886142, 4.07207971681606651436400388022, 5.12043966050229929483172329963, 6.12300047178091251100540461899, 7.47980166778831526144096155091, 8.183830952219833514625348454369, 9.481567249528061317114778608345, 9.925476831639531708752742679800, 11.17587476422908202359783295076, 12.88556162027869904406527449170, 14.08547434157835425830924658942

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