Properties

Label 2-63e2-441.281-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.484 - 0.874i$
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 + (0.955 + 0.294i)7-s + (−0.914 + 0.848i)13-s + (0.365 + 0.930i)16-s − 0.445·19-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)28-s + (0.900 + 1.56i)31-s + (1.62 + 0.781i)37-s + (−0.162 − 0.414i)43-s + (0.826 + 0.563i)49-s + (−1.23 + 0.185i)52-s + (−1.48 + 1.01i)61-s + (−0.222 + 0.974i)64-s + (−0.623 − 1.07i)67-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)4-s + (0.955 + 0.294i)7-s + (−0.914 + 0.848i)13-s + (0.365 + 0.930i)16-s − 0.445·19-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)28-s + (0.900 + 1.56i)31-s + (1.62 + 0.781i)37-s + (−0.162 − 0.414i)43-s + (0.826 + 0.563i)49-s + (−1.23 + 0.185i)52-s + (−1.48 + 1.01i)61-s + (−0.222 + 0.974i)64-s + (−0.623 − 1.07i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.484 - 0.874i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2780, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.680383711\)
\(L(\frac12)\) \(\approx\) \(1.680383711\)
\(L(1)\) 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 \( 1 + (-0.955 - 0.294i)T \)
good2 \( 1 + (-0.826 - 0.563i)T^{2} \)
5 \( 1 + (0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.914 - 0.848i)T + (0.0747 - 0.997i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + (-0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.365 + 0.930i)T^{2} \)
31 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.733 + 0.680i)T^{2} \)
43 \( 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (1.48 - 1.01i)T + (0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.0747 - 0.997i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.610712940590379935381470702537, −7.931882049248359349699389692348, −7.42257691253784613452633359571, −6.56193826719560591624652370859, −5.99659032540875109196718762082, −4.80047368457255530789184377633, −4.37049643726258671880468750489, −3.15837778267057541911604987499, −2.32826139432958307586346591691, −1.59580956207484026405939527494, 0.949386490301175471173429566257, 2.09918340397713015113688968948, 2.73632884031631028325568893189, 4.01272057745953146598971549862, 4.83855260639899136427776090735, 5.60507956240366854147231468009, 6.20069105832742116965232851931, 7.19207229406722334037335730094, 7.74211281637453408004033229149, 8.217828924927580764271813595264

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