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Properties

Label 2-3381-483.482-c0-0-13
Degree $2$
Conductor $3381$
Sign $0.972 - 0.230i$
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  − 0.517i·2-s + (0.608 + 0.793i)3-s + 0.732·4-s + (0.410 − 0.315i)6-s − 0.896i·8-s + (−0.258 + 0.965i)9-s + (0.445 + 0.580i)12-s + 0.261i·13-s + 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (0.711 − 0.545i)24-s + 25-s + 0.135·26-s + (−0.923 + 0.382i)27-s + ⋯
L(s)  = 1  − 0.517i·2-s + (0.608 + 0.793i)3-s + 0.732·4-s + (0.410 − 0.315i)6-s − 0.896i·8-s + (−0.258 + 0.965i)9-s + (0.445 + 0.580i)12-s + 0.261i·13-s + 0.267·16-s + (0.499 + 0.133i)18-s + i·23-s + (0.711 − 0.545i)24-s + 25-s + 0.135·26-s + (−0.923 + 0.382i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.972 - 0.230i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ 0.972 - 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.938186673\)
\(L(\frac12)\) \(\approx\) \(1.938186673\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 \)
23 \( 1 - iT \)
good2 \( 1 + 0.517iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 0.261iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 - 1.98iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.261T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.58T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + 1.21iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
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

−9.049532152909666910231987448221, −8.118188487245616446912968893664, −7.43363103940815600521451997291, −6.65445714905434424520304206646, −5.73530130652868569862568749258, −4.83414580946017920591173116440, −3.94532648076207821645692886391, −3.20220530350896087899701655620, −2.48109238878667777501257387840, −1.48051666474246412088826722534, 1.20163804158237811100065710166, 2.35297937515795105203320349116, 2.91688969614217924554021589490, 4.01078743699082293371783627531, 5.22021595997896132657736004355, 5.99682157406359439627614131507, 6.70854298419358473075171964098, 7.22916493286040916018384223500, 7.907612092293379108879857737348, 8.569962880470377719738176380418

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