↑

Properties

Label 2-3381-483.206-c0-0-13
Degree $2$
Conductor $3381$
Sign $0.864 + 0.503i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.864 + 0.503i$
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: \((2,\ 3381,\ (\ :0),\ 0.864 + 0.503i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4609372768\)
\(L(\frac12)\) \(\approx\) \(0.4609372768\)
\(L(1)\) 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 \( 1 + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.764897725840341659771055677005, −7.88179993265138107317638898304, −7.25716168426396765898190497783, −6.61426517640676000051145494985, −5.76393336967834234896116558015, −4.94649862987667815351842454561, −4.11420493679097386565642009305, −3.45131905685072103240445252798, −1.97452320937892294480365294559, −0.44466975381725914163061246873, 1.06438364112269387695717614970, 1.91936549095201866631795406670, 3.32093305065972117952805694677, 4.48593536728744237241566061285, 5.20744527527294914055304153627, 5.68234899989717813974560924868, 6.60378850049431535496780361551, 7.27982645907152126743209004458, 8.195393898501209536466100060836, 9.045035331806329397652648979419

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