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Properties

Label 2-63e2-441.13-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.777 + 0.629i$
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  + (−0.0747 + 0.997i)4-s + (0.733 − 0.680i)7-s + (0.460 − 1.49i)13-s + (−0.988 − 0.149i)16-s − 1.94i·19-s + (0.955 − 0.294i)25-s + (0.623 + 0.781i)28-s + (−0.751 − 0.433i)31-s + (−1.62 − 0.781i)37-s + (−0.440 − 0.0663i)43-s + (0.0747 − 0.997i)49-s + (1.45 + 0.571i)52-s + (0.865 − 0.0648i)61-s + (0.222 − 0.974i)64-s + (−0.623 + 1.07i)67-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)4-s + (0.733 − 0.680i)7-s + (0.460 − 1.49i)13-s + (−0.988 − 0.149i)16-s − 1.94i·19-s + (0.955 − 0.294i)25-s + (0.623 + 0.781i)28-s + (−0.751 − 0.433i)31-s + (−1.62 − 0.781i)37-s + (−0.440 − 0.0663i)43-s + (0.0747 − 0.997i)49-s + (1.45 + 0.571i)52-s + (0.865 − 0.0648i)61-s + (0.222 − 0.974i)64-s + (−0.623 + 1.07i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.777 + 0.629i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.777 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295758497\)
\(L(\frac12)\) \(\approx\) \(1.295758497\)
\(L(1)\) 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 \( 1 + (0.0747 - 0.997i)T^{2} \)
5 \( 1 + (-0.955 + 0.294i)T^{2} \)
11 \( 1 + (0.0747 - 0.997i)T^{2} \)
13 \( 1 + (-0.460 + 1.49i)T + (-0.826 - 0.563i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + 1.94iT - T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.988 + 0.149i)T^{2} \)
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 \( 1 + (-0.955 + 0.294i)T^{2} \)
43 \( 1 + (0.440 + 0.0663i)T + (0.955 + 0.294i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (-0.865 + 0.0648i)T + (0.988 - 0.149i)T^{2} \)
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 \( 1 + (-0.826 + 0.563i)T^{2} \)
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

−8.682922526035129004690801990832, −7.73174308381252127253296709085, −7.27652061335467948506388824531, −6.59020351308781307395878651869, −5.34280751938191737077792149953, −4.80812625433685322442232667062, −3.87411960679815572766800825782, −3.16571741695049057595577147552, −2.27063909211652052515849869993, −0.75011421858192475610822140725, 1.60588728365958744546395482766, 1.81352260121021861841004230859, 3.31778199741307104351650883878, 4.32225913836731625358367780039, 5.05066652885608245989476431848, 5.71806818584263200417606806193, 6.41549752325740012394530109490, 7.11165258754750744284109968793, 8.154617252370657089432580760968, 8.829225278938666226263946222472

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