Properties

Label 2-3381-483.482-c0-0-5
Degree $2$
Conductor $3381$
Sign $-0.474 - 0.880i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.474 - 0.880i$
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: \((2,\ 3381,\ (\ :0),\ -0.474 - 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230163172\)
\(L(\frac12)\) \(\approx\) \(1.230163172\)
\(L(1)\) 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

−8.784662800760318743432204517755, −8.514802709911770103951391926470, −7.25410567769072860055569874922, −6.72047933601997219166864040845, −6.16837939610578988903606094946, −5.11930242088745407209621687699, −4.83629761710723857283495595502, −3.53499511585774534729600922854, −2.80878528002281230797155416319, −1.41564203404237005196404403829, 0.822766434379721603333092941062, 1.93720979162199527097191805763, 2.64555491591659331935201148681, 3.69044964697093116059559697178, 4.75791341722479969192516981708, 5.80911595638999091095668446108, 6.22729950196543297586732963650, 7.09000122051622853126416159892, 7.67930156968148501547320148644, 8.303714618774464678427308332915

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