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Properties

Label 2-3381-483.206-c0-0-5
Degree $2$
Conductor $3381$
Sign $-0.846 - 0.532i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.846 - 0.532i$
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: \((2,\ 3381,\ (\ :0),\ -0.846 - 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.197980574\)
\(L(\frac12)\) \(\approx\) \(1.197980574\)
\(L(1)\) 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

−9.159952814043530843106265413658, −8.552550786176538457083765895111, −7.76031801853875734675656302955, −6.83848037205873214308720692631, −5.91060940227302754250952379970, −4.93336054261563854969567091590, −4.39273712989226925718557802603, −3.80999257729347236750469675419, −2.92278109079993705673178699003, −2.00084414408003877445382540578, 0.60069759376096798417234837499, 1.67507276427432205730795824515, 2.98225401546092879402580805054, 3.63788309980922857987411544322, 5.14965484438641173278987186319, 5.29961355059057464044479695198, 6.27420697024961028780832634147, 6.86418408743553973181558786169, 7.68446684980970374745722773081, 8.386322724740801349430881528300

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