Properties

Label 2-3381-483.206-c0-0-16
Degree $2$
Conductor $3381$
Sign $-0.292 + 0.956i$
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 + (−0.130 − 0.991i)3-s + (−0.366 + 0.633i)4-s + (−0.315 − 0.410i)6-s + 0.896i·8-s + (−0.965 + 0.258i)9-s + (0.676 + 0.280i)12-s − 1.98i·13-s + (−0.133 − 0.232i)16-s + (−0.366 + 0.366i)18-s + (0.866 − 0.5i)23-s + (0.888 − 0.117i)24-s + (−0.5 + 0.866i)25-s + (−0.513 − 0.888i)26-s + (0.382 + 0.923i)27-s + ⋯
L(s)  = 1  + (0.448 − 0.258i)2-s + (−0.130 − 0.991i)3-s + (−0.366 + 0.633i)4-s + (−0.315 − 0.410i)6-s + 0.896i·8-s + (−0.965 + 0.258i)9-s + (0.676 + 0.280i)12-s − 1.98i·13-s + (−0.133 − 0.232i)16-s + (−0.366 + 0.366i)18-s + (0.866 − 0.5i)23-s + (0.888 − 0.117i)24-s + (−0.5 + 0.866i)25-s + (−0.513 − 0.888i)26-s + (0.382 + 0.923i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.292 + 0.956i$
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: \((2,\ 3381,\ (\ :0),\ -0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148142885\)
\(L(\frac12)\) \(\approx\) \(1.148142885\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.130 + 0.991i)T \)
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 \( 1 + 1.98iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (0.226 + 0.130i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.98T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.608 + 1.05i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \)
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

−8.320798502227600986110702805353, −7.79971659168551369791714597607, −7.33362345617339580061337350461, −6.17821005083481242296078130737, −5.54309202249329935804090598960, −4.84504075214634462106519257059, −3.68201346101314523659624738400, −2.94515843551750566362703096118, −2.19188825418740082496503027251, −0.63212736159198824484087811315, 1.40910150004471379329074796041, 2.80076541761350109971376786264, 4.00876914007950442448909899036, 4.33524659816123356413764119411, 5.14764336031779768178828542982, 5.85065483499880024516525397540, 6.60054782972992240217075514595, 7.28492275760284670313727526813, 8.634321160124740785989360010196, 9.130179126995206421928119338226

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