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Properties

Label 2-63e2-441.283-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 + (−0.548 + 1.77i)13-s + (0.365 + 0.930i)16-s + (−1.68 + 0.974i)19-s + (−0.222 + 0.974i)25-s + (−0.988 + 0.149i)28-s + (−1.72 + 0.997i)31-s + (−0.0111 − 0.149i)37-s + (1.88 + 0.284i)43-s + (0.0747 − 0.997i)49-s + (1.45 − 1.16i)52-s + (−0.634 − 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 + (−0.548 + 1.77i)13-s + (0.365 + 0.930i)16-s + (−1.68 + 0.974i)19-s + (−0.222 + 0.974i)25-s + (−0.988 + 0.149i)28-s + (−1.72 + 0.997i)31-s + (−0.0111 − 0.149i)37-s + (1.88 + 0.284i)43-s + (0.0747 − 0.997i)49-s + (1.45 − 1.16i)52-s + (−0.634 − 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: $\chi_{3969} (2782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.156 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7070736129\)
\(L(\frac12)\) \(\approx\) \(0.7070736129\)
\(L(1)\) 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 \( 1 + (0.548 - 1.77i)T + (-0.826 - 0.563i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.988 + 0.149i)T^{2} \)
31 \( 1 + (1.72 - 0.997i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.634 + 0.930i)T + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.590 + 0.636i)T + (-0.0747 - 0.997i)T^{2} \)
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 \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.898979104187966264994258691460, −8.168750383457153295676704713968, −7.30695505173448683512374666206, −6.65166648832846160661141679714, −5.72583956002042645031337516156, −4.97607449812258945588712281220, −4.15487250057929968041000197827, −3.86905793139905928610812227962, −2.08727851187436172123223841588, −1.43273452004547430724179965067, 0.40642214940556207912272534927, 2.20162142737553422307638610213, 2.88201703983541939648405821673, 4.01295893249536489892864337893, 4.66708286220936518198299218439, 5.42919204625817911875624095498, 6.03985938553685249141551027999, 7.28876812114102865956090310954, 7.87593751711261304374451374631, 8.447810363261819957185794373965

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