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Properties

Label 2-3879-3879.430-c0-0-11
Degree $2$
Conductor $3879$
Sign $0.623 - 0.781i$
Analytic cond. $1.93587$
Root an. cond. $1.39135$
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  + (−0.623 + 1.07i)2-s + (0.698 − 0.715i)3-s + (−0.277 − 0.480i)4-s + (0.661 + 1.14i)5-s + (0.337 + 1.20i)6-s − 0.554·8-s + (−0.0249 − 0.999i)9-s − 1.65·10-s + (0.939 − 1.62i)11-s + (−0.537 − 0.136i)12-s + (1.28 + 0.326i)15-s + (0.623 − 1.07i)16-s + (1.09 + 0.596i)18-s − 0.822·19-s + (0.367 − 0.636i)20-s + ⋯
L(s)  = 1  + (−0.623 + 1.07i)2-s + (0.698 − 0.715i)3-s + (−0.277 − 0.480i)4-s + (0.661 + 1.14i)5-s + (0.337 + 1.20i)6-s − 0.554·8-s + (−0.0249 − 0.999i)9-s − 1.65·10-s + (0.939 − 1.62i)11-s + (−0.537 − 0.136i)12-s + (1.28 + 0.326i)15-s + (0.623 − 1.07i)16-s + (1.09 + 0.596i)18-s − 0.822·19-s + (0.367 − 0.636i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3879\)    =    \(3^{2} \cdot 431\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(1.93587\)
Root analytic conductor: \(1.39135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3879} (430, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3879,\ (\ :0),\ 0.623 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400139458\)
\(L(\frac12)\) \(\approx\) \(1.400139458\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.698 + 0.715i)T \)
431 \( 1 - T \)
good2 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.661 - 1.14i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.822T + T^{2} \)
23 \( 1 + (-0.797 - 1.38i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.998 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.99T + T^{2} \)
59 \( 1 + (-0.0249 - 0.0431i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.456 - 0.790i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.540971663361524717215710050276, −8.048914913563766951507693102989, −7.22485785251199047566748600772, −6.62427714167501359450511590742, −6.16839345751263514946285422531, −5.68116864774676770029761365392, −3.94335203012628453038621119080, −3.09083447851487265259439035413, −2.50071450332659200801616935297, −1.06766235063378593130220316067, 1.25128009975846939969672768826, 2.00250646694189818592045976165, 2.73033131752101680678757160910, 3.90796844453090093321380722420, 4.60286071528637309266818864463, 5.20815066666204106696119951915, 6.37608887078145876343847062715, 7.21023599683609039027868846878, 8.454677565197246328815966687184, 8.842598093812874267934976264050

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