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Properties

Label 2-3789-421.59-c0-0-0
Degree $2$
Conductor $3789$
Sign $0.987 - 0.159i$
Analytic cond. $1.89095$
Root an. cond. $1.37512$
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  + (0.657 + 0.753i)4-s + (0.988 − 1.65i)7-s + (0.482 + 0.0764i)13-s + (−0.134 + 0.990i)16-s + (−0.931 + 1.64i)19-s + (0.134 + 0.990i)25-s + (1.89 − 0.344i)28-s + (0.543 − 0.568i)31-s + (0.571 − 0.477i)37-s + (0.644 + 0.846i)43-s + (−1.28 − 2.39i)49-s + (0.259 + 0.413i)52-s + (1.22 − 0.930i)61-s + (−0.834 + 0.550i)64-s + (1.17 − 1.61i)67-s + ⋯
L(s)  = 1  + (0.657 + 0.753i)4-s + (0.988 − 1.65i)7-s + (0.482 + 0.0764i)13-s + (−0.134 + 0.990i)16-s + (−0.931 + 1.64i)19-s + (0.134 + 0.990i)25-s + (1.89 − 0.344i)28-s + (0.543 − 0.568i)31-s + (0.571 − 0.477i)37-s + (0.644 + 0.846i)43-s + (−1.28 − 2.39i)49-s + (0.259 + 0.413i)52-s + (1.22 − 0.930i)61-s + (−0.834 + 0.550i)64-s + (1.17 − 1.61i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735711514\)
\(L(\frac12)\) \(\approx\) \(1.735711514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
421 \( 1 + (-0.512 - 0.858i)T \)
good2 \( 1 + (-0.657 - 0.753i)T^{2} \)
5 \( 1 + (-0.134 - 0.990i)T^{2} \)
7 \( 1 + (-0.988 + 1.65i)T + (-0.473 - 0.880i)T^{2} \)
11 \( 1 + (0.983 + 0.178i)T^{2} \)
13 \( 1 + (-0.482 - 0.0764i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.393 - 0.919i)T^{2} \)
19 \( 1 + (0.931 - 1.64i)T + (-0.512 - 0.858i)T^{2} \)
23 \( 1 + (0.722 - 0.691i)T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (-0.543 + 0.568i)T + (-0.0448 - 0.998i)T^{2} \)
37 \( 1 + (-0.571 + 0.477i)T + (0.178 - 0.983i)T^{2} \)
41 \( 1 + (0.512 + 0.858i)T^{2} \)
43 \( 1 + (-0.644 - 0.846i)T + (-0.266 + 0.963i)T^{2} \)
47 \( 1 + (0.990 - 0.134i)T^{2} \)
53 \( 1 + (0.880 - 0.473i)T^{2} \)
59 \( 1 + (-0.0896 - 0.995i)T^{2} \)
61 \( 1 + (-1.22 + 0.930i)T + (0.266 - 0.963i)T^{2} \)
67 \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.0896 + 0.995i)T^{2} \)
73 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.0943 - 0.696i)T + (-0.963 + 0.266i)T^{2} \)
83 \( 1 + (0.0896 + 0.995i)T^{2} \)
89 \( 1 + (0.998 + 0.0448i)T^{2} \)
97 \( 1 + (0.344 + 0.0950i)T + (0.858 + 0.512i)T^{2} \)
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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

−8.195712533997402510540348480234, −8.066968307648064386766155184543, −7.34206394680504974897841940371, −6.63891419463617787360179891680, −5.87938097668542708885705994163, −4.68640393076462309690336912828, −3.96126852169463096209693040131, −3.48690244399162253313673818755, −2.11640430994794611764729296108, −1.27964716885780214534915884086, 1.21378466845557942886618114775, 2.45102194924427064958984928776, 2.58186210836856567546863771370, 4.28880263240367517459468706992, 5.08426022606467107413783177103, 5.64231662377639908279476446079, 6.39516513245557241113655301564, 7.00385888664231866303857610286, 8.131044559012427554738885984814, 8.663078197975634889016330898940

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