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Properties

Label 2-63e2-441.254-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.156 + 0.987i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
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  + (0.826 − 0.563i)4-s + (−0.733 − 0.680i)7-s + (0.698 − 0.215i)13-s + (0.365 − 0.930i)16-s + (0.222 − 0.385i)19-s + (−0.222 − 0.974i)25-s + (−0.988 − 0.149i)28-s + (−0.0747 + 0.129i)31-s + (0.0111 − 0.149i)37-s + (−1.88 + 0.284i)43-s + (0.0747 + 0.997i)49-s + (0.455 − 0.571i)52-s + (1.36 + 0.930i)61-s + (−0.222 − 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)4-s + (−0.733 − 0.680i)7-s + (0.698 − 0.215i)13-s + (0.365 − 0.930i)16-s + (0.222 − 0.385i)19-s + (−0.222 − 0.974i)25-s + (−0.988 − 0.149i)28-s + (−0.0747 + 0.129i)31-s + (0.0111 − 0.149i)37-s + (−1.88 + 0.284i)43-s + (0.0747 + 0.997i)49-s + (0.455 − 0.571i)52-s + (1.36 + 0.930i)61-s + (−0.222 − 0.974i)64-s + (0.988 − 1.71i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425928776\)
\(L(\frac12)\) \(\approx\) \(1.425928776\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.733 + 0.680i)T \)
good2 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.698 + 0.215i)T + (0.826 - 0.563i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.988 + 0.149i)T^{2} \)
31 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.0111 + 0.149i)T + (-0.988 - 0.149i)T^{2} \)
41 \( 1 + (-0.955 - 0.294i)T^{2} \)
43 \( 1 + (1.88 - 0.284i)T + (0.955 - 0.294i)T^{2} \)
47 \( 1 + (-0.826 + 0.563i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-1.32 + 1.22i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.826 - 0.563i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.366265297125185418118279659046, −7.66283071425902987344552345322, −6.76529353384394091280501135757, −6.46530689190417406809974299715, −5.63522446104333279891213279085, −4.76700725196998507427877058408, −3.70494977627230099051346379906, −2.99662658820051653722808348029, −1.94741511821111316387372172183, −0.790108178854238334481285065671, 1.56132440204366470410338398740, 2.50796884974893632332655300557, 3.37716597517902326384118583293, 3.91516541277305895367203228033, 5.29387520789798336311375807069, 5.89728605839153575945188422023, 6.74576277232171723435338054487, 7.11605500833150325232936463066, 8.291714870575548316808975806670, 8.500657232691659935550530247295

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