Properties

Label 2-3381-483.206-c0-0-7
Degree $2$
Conductor $3381$
Sign $-0.681 - 0.731i$
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  + (−1.67 + 0.965i)2-s + (0.923 + 0.382i)3-s + (1.36 − 2.36i)4-s + (−1.91 + 0.252i)6-s + 3.34i·8-s + (0.707 + 0.707i)9-s + (2.16 − 1.66i)12-s + 1.21i·13-s + (−1.86 − 3.23i)16-s + (−1.86 − 0.500i)18-s + (0.866 − 0.5i)23-s + (−1.28 + 3.09i)24-s + (−0.5 + 0.866i)25-s + (−1.17 − 2.03i)26-s + (0.382 + 0.923i)27-s + ⋯
L(s)  = 1  + (−1.67 + 0.965i)2-s + (0.923 + 0.382i)3-s + (1.36 − 2.36i)4-s + (−1.91 + 0.252i)6-s + 3.34i·8-s + (0.707 + 0.707i)9-s + (2.16 − 1.66i)12-s + 1.21i·13-s + (−1.86 − 3.23i)16-s + (−1.86 − 0.500i)18-s + (0.866 − 0.5i)23-s + (−1.28 + 3.09i)24-s + (−0.5 + 0.866i)25-s + (−1.17 − 2.03i)26-s + (0.382 + 0.923i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.681 - 0.731i$
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: \((2,\ 3381,\ (\ :0),\ -0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7468102230\)
\(L(\frac12)\) \(\approx\) \(0.7468102230\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 \)
23 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.21iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.21T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.991 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + (0.226 + 0.130i)T + (0.5 + 0.866i)T^{2} \)
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

−9.033135947118051662473557197909, −8.514403420455424226777221902012, −7.61565380409109016913960285251, −7.16029475205670702897254099704, −6.51923460784085605612372806093, −5.46250654914457962280050202226, −4.69661381766015896578696424030, −3.42840255192123573553781512385, −2.17973426384784257740296934425, −1.45584176852012767858072463921, 0.70443945818748904086687883620, 1.82243584791807197715550160704, 2.60686341536080900306497530776, 3.34000181105348966722699484201, 4.08250889379916650209150960601, 5.69012309578300358416527286225, 6.92162232407880715432377374589, 7.33105979218310085041890016008, 8.149330880079050383195675440611, 8.514480020986570596454692006091

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