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Properties

Label 2-3381-483.206-c0-0-14
Degree $2$
Conductor $3381$
Sign $0.893 - 0.448i$
Analytic cond. $1.68733$
Root an. cond. $1.29897$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.448 + 0.258i)2-s + (0.991 − 0.130i)3-s + (−0.366 + 0.633i)4-s + (−0.410 + 0.315i)6-s − 0.896i·8-s + (0.965 − 0.258i)9-s + (−0.280 + 0.676i)12-s − 0.261i·13-s + (−0.133 − 0.232i)16-s + (−0.366 + 0.366i)18-s + (0.866 − 0.5i)23-s + (−0.117 − 0.888i)24-s + (−0.5 + 0.866i)25-s + (0.0675 + 0.117i)26-s + (0.923 − 0.382i)27-s + ⋯
L(s)  = 1  + (−0.448 + 0.258i)2-s + (0.991 − 0.130i)3-s + (−0.366 + 0.633i)4-s + (−0.410 + 0.315i)6-s − 0.896i·8-s + (0.965 − 0.258i)9-s + (−0.280 + 0.676i)12-s − 0.261i·13-s + (−0.133 − 0.232i)16-s + (−0.366 + 0.366i)18-s + (0.866 − 0.5i)23-s + (−0.117 − 0.888i)24-s + (−0.5 + 0.866i)25-s + (0.0675 + 0.117i)26-s + (0.923 − 0.382i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (2138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.351345215\)
\(L(\frac12)\) \(\approx\) \(1.351345215\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.991 + 0.130i)T \)
7 \( 1 \)
23 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 0.261iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-1.71 - 0.991i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.261T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + (1.05 + 0.608i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.821234154927325267271882289346, −8.042095057804886736191983085406, −7.66230909838161899428725376236, −6.88660130249867100832016156018, −6.10264636489131755744422675986, −4.75939446403191298225970371706, −4.14311838116112483507242707832, −3.20426985538322206912489902443, −2.55647119443531271044375147476, −1.10268542960090538979476669183, 1.13773519763976226215685941757, 2.13599766547962100765621859515, 2.99534513642764571921355228291, 4.08633552211723413928099970693, 4.78214135578405402970240641332, 5.63975932678056223572811295699, 6.64469635196933822030057804119, 7.41509953584164061239199134887, 8.357460140303079398495474705285, 8.698923135955910539317237260270

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