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Properties

Label 2-165-165.164-c1-0-12
Degree $2$
Conductor $165$
Sign $0.0893 + 0.995i$
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  − 2.39i·2-s + (0.796 + 1.53i)3-s − 3.73·4-s + (2.17 − 0.517i)5-s + (3.68 − 1.90i)6-s + 3.38·7-s + 4.14i·8-s + (−1.73 + 2.44i)9-s + (−1.23 − 5.20i)10-s + (−2.69 − 1.93i)11-s + (−2.97 − 5.74i)12-s − 3.38·13-s − 8.10i·14-s + (2.52 + 2.93i)15-s + 2.46·16-s − 1.75i·17-s + ⋯
L(s)  = 1  − 1.69i·2-s + (0.459 + 0.888i)3-s − 1.86·4-s + (0.972 − 0.231i)5-s + (1.50 − 0.778i)6-s + 1.27·7-s + 1.46i·8-s + (−0.577 + 0.816i)9-s + (−0.391 − 1.64i)10-s + (−0.812 − 0.582i)11-s + (−0.857 − 1.65i)12-s − 0.939·13-s − 2.16i·14-s + (0.652 + 0.757i)15-s + 0.616·16-s − 0.425i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0893 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0893 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.0893 + 0.995i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.0893 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02799 - 0.939871i\)
\(L(\frac12)\) \(\approx\) \(1.02799 - 0.939871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.796 - 1.53i)T \)
5 \( 1 + (-2.17 + 0.517i)T \)
11 \( 1 + (2.69 + 1.93i)T \)
good2 \( 1 + 2.39iT - 2T^{2} \)
7 \( 1 - 3.38T + 7T^{2} \)
13 \( 1 + 3.38T + 13T^{2} \)
17 \( 1 + 1.75iT - 17T^{2} \)
19 \( 1 - 3.81iT - 19T^{2} \)
23 \( 1 - 2.75T + 23T^{2} \)
29 \( 1 + 1.97T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 8.40iT - 37T^{2} \)
41 \( 1 + 7.36T + 41T^{2} \)
43 \( 1 - 8.34T + 43T^{2} \)
47 \( 1 - 1.59T + 47T^{2} \)
53 \( 1 + 8.70T + 53T^{2} \)
59 \( 1 + 4.89iT - 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 + 3.58iT - 71T^{2} \)
73 \( 1 - 3.38T + 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 - 9.52iT - 89T^{2} \)
97 \( 1 + 8.40iT - 97T^{2} \)
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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

−12.44749938786088443232514429457, −11.31265806234764090989994951146, −10.60580778945530852697392735580, −9.851172936666407135845396664320, −8.988098085371758248538345515376, −7.979754979308382048834494964464, −5.32860408911511257746447191372, −4.65504112399937767408159970677, −3.04370238687594301871150349531, −1.89919815695034384526443811468, 2.20932985614766191360434897532, 4.86195652504329733113369411575, 5.71099722216046855525356467729, 7.02783562172983915887883066120, 7.55772797302759205653796301821, 8.592164788774499536685004118447, 9.508883159404749279988088529913, 11.02761764636609803365049949654, 12.63842248666301220386111410846, 13.38894158367301699305255339293

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