Properties

Label 2-63e2-441.241-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.772 - 0.634i$
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  + (−0.623 + 0.781i)4-s + (−0.0747 + 0.997i)7-s + (−0.332 + 0.487i)13-s + (−0.222 − 0.974i)16-s + (0.751 − 0.433i)19-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)28-s + 1.86i·31-s + (−0.266 + 0.680i)37-s + (−0.142 − 0.0440i)43-s + (−0.988 − 0.149i)49-s + (−0.173 − 0.563i)52-s + (−0.233 + 0.185i)61-s + (0.900 + 0.433i)64-s − 1.46·67-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)4-s + (−0.0747 + 0.997i)7-s + (−0.332 + 0.487i)13-s + (−0.222 − 0.974i)16-s + (0.751 − 0.433i)19-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)28-s + 1.86i·31-s + (−0.266 + 0.680i)37-s + (−0.142 − 0.0440i)43-s + (−0.988 − 0.149i)49-s + (−0.173 − 0.563i)52-s + (−0.233 + 0.185i)61-s + (0.900 + 0.433i)64-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.772 - 0.634i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.772 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7937859153\)
\(L(\frac12)\) \(\approx\) \(0.7937859153\)
\(L(1)\) 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 \( 1 + (0.0747 - 0.997i)T \)
good2 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.332 - 0.487i)T + (-0.365 - 0.930i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.751 + 0.433i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.955 - 0.294i)T^{2} \)
31 \( 1 - 1.86iT - T^{2} \)
37 \( 1 + (0.266 - 0.680i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (-0.826 + 0.563i)T^{2} \)
43 \( 1 + (0.142 + 0.0440i)T + (0.826 + 0.563i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (0.233 - 0.185i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + 1.46T + T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.880 + 1.29i)T + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + 1.97T + T^{2} \)
83 \( 1 + (-0.365 + 0.930i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.869286686397118846680660611882, −8.391629661034834080784917758554, −7.41285465380690969359637528741, −6.93140410916061384363173715023, −5.84912698203554851423471923449, −5.06782564939655650218180224034, −4.51624038539327086048133120392, −3.31603775170450273865556345571, −2.88620722731986814093043081617, −1.59692863632769010228826221830, 0.46481250494549227139327711678, 1.57085427512501838837208577567, 2.84191505310611556549795960090, 3.99360223446563547380497704979, 4.43927202102012172859120077416, 5.43789814085618133389667554143, 5.98716848839833685312327855156, 6.89965937458739291633636111396, 7.66566260834003190240864092330, 8.291820040633440525184172272355

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