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Properties

Label 2-165-33.29-c1-0-8
Degree $2$
Conductor $165$
Sign $0.491 + 0.870i$
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  + (0.131 + 0.0958i)2-s + (−1.46 + 0.923i)3-s + (−0.609 − 1.87i)4-s + (−0.587 − 0.809i)5-s + (−0.281 − 0.0186i)6-s + (3.41 − 1.10i)7-s + (0.200 − 0.616i)8-s + (1.29 − 2.70i)9-s − 0.163i·10-s + (−2.00 − 2.64i)11-s + (2.62 + 2.18i)12-s + (1.34 − 1.84i)13-s + (0.556 + 0.180i)14-s + (1.60 + 0.642i)15-s + (−3.10 + 2.25i)16-s + (2.62 − 1.90i)17-s + ⋯
L(s)  = 1  + (0.0932 + 0.0677i)2-s + (−0.846 + 0.533i)3-s + (−0.304 − 0.938i)4-s + (−0.262 − 0.361i)5-s + (−0.115 − 0.00761i)6-s + (1.29 − 0.419i)7-s + (0.0707 − 0.217i)8-s + (0.431 − 0.902i)9-s − 0.0515i·10-s + (−0.604 − 0.796i)11-s + (0.758 + 0.631i)12-s + (0.372 − 0.512i)13-s + (0.148 + 0.0483i)14-s + (0.415 + 0.165i)15-s + (−0.776 + 0.564i)16-s + (0.635 − 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.491 + 0.870i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.491 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762090 - 0.444928i\)
\(L(\frac12)\) \(\approx\) \(0.762090 - 0.444928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.46 - 0.923i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (2.00 + 2.64i)T \)
good2 \( 1 + (-0.131 - 0.0958i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (-3.41 + 1.10i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.34 + 1.84i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.62 + 1.90i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.12 + 0.365i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 + (-1.32 - 4.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.54 + 2.57i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.98 + 6.10i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.76 + 5.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.38iT - 43T^{2} \)
47 \( 1 + (-10.6 - 3.44i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.65 - 2.28i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-9.33 + 3.03i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.91 - 6.76i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.67T + 67T^{2} \)
71 \( 1 + (-2.26 - 3.11i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-11.2 + 3.65i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.48 - 7.54i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.6 - 9.88i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9.77iT - 89T^{2} \)
97 \( 1 + (-4.81 - 3.49i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.61100769367563361955959135031, −11.18672736107518338494908641701, −10.96384522903196866876575872025, −9.836222048229667464947857314637, −8.669616569616226606281501885479, −7.39698931721845704516193409727, −5.68425233220828953014672440104, −5.21307227047110788405580143495, −4.00062042629488867835737747058, −1.00424872401269365309374877554, 2.18936789549670015302952284619, 4.25426294341652631657679057335, 5.26006506943975932687949018862, 6.80998927375525528927492328092, 7.82949088919717970149038048290, 8.512782736975498518033052058579, 10.30702486839230188744157107671, 11.29668199952414670650656701455, 12.04550556728386479196785594745, 12.69278143685010727308866648338

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