Properties

Label 2-63e2-441.328-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 + (1.06 − 1.14i)13-s + (0.365 − 0.930i)16-s + 1.94i·19-s + (−0.733 + 0.680i)25-s + (0.623 − 0.781i)28-s + (0.751 + 0.433i)31-s + (−1.62 + 0.781i)37-s + (0.162 − 0.414i)43-s + (0.826 − 0.563i)49-s + (−0.233 + 1.54i)52-s + (−0.488 + 0.716i)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 + (1.06 − 1.14i)13-s + (0.365 − 0.930i)16-s + 1.94i·19-s + (−0.733 + 0.680i)25-s + (0.623 − 0.781i)28-s + (0.751 + 0.433i)31-s + (−1.62 + 0.781i)37-s + (0.162 − 0.414i)43-s + (0.826 − 0.563i)49-s + (−0.233 + 1.54i)52-s + (−0.488 + 0.716i)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: $\chi_{3969} (3268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6416787420\)
\(L(\frac12)\) \(\approx\) \(0.6416787420\)
\(L(1)\) 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 \( 1 + (-1.06 + 1.14i)T + (-0.0747 - 0.997i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 - 1.94iT - T^{2} \)
23 \( 1 + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (0.365 + 0.930i)T^{2} \)
31 \( 1 + (-0.751 - 0.433i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.488 - 0.716i)T + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \)
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 \( 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.681656951964383069997229645418, −8.301640750561655982188785407539, −7.59226968167809478180585901925, −6.64072591538172823591654787391, −5.74227459904348139491474739621, −5.36513338838092815907224277861, −4.03370853233687819902553242732, −3.55230556418608911108931346684, −2.87036934236941822547148360105, −1.31842647763560610763178888034, 0.39860510380244530895529165259, 1.72189017989836671064417184500, 2.97515063583304998174439084369, 3.96857681654236469610954597660, 4.47574037842102275984902667852, 5.40400741698655495493659154660, 6.36637436876322281362007716241, 6.62857447536710351466643547776, 7.66702296599051633927881291135, 8.718090302485110364383064615981

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