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Properties

Label 2-63e2-441.250-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.851 - 0.524i$
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  + (0.733 + 0.680i)4-s + (0.988 − 0.149i)7-s + (1.85 + 0.728i)13-s + (0.0747 + 0.997i)16-s + (−1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.826 + 0.563i)28-s + (0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (−0.603 + 0.411i)43-s + (0.955 − 0.294i)49-s + (0.865 + 1.79i)52-s + (−0.925 − 0.997i)61-s + (−0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)4-s + (0.988 − 0.149i)7-s + (1.85 + 0.728i)13-s + (0.0747 + 0.997i)16-s + (−1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.826 + 0.563i)28-s + (0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (−0.603 + 0.411i)43-s + (0.955 − 0.294i)49-s + (0.865 + 1.79i)52-s + (−0.925 − 0.997i)61-s + (−0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.859378536\)
\(L(\frac12)\) \(\approx\) \(1.859378536\)
\(L(1)\) 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 \( 1 + (-0.988 + 0.149i)T \)
good2 \( 1 + (-0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (-1.85 - 0.728i)T + (0.733 + 0.680i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.826 + 0.563i)T^{2} \)
31 \( 1 + (-0.510 - 0.294i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.82 + 0.563i)T + (0.826 + 0.563i)T^{2} \)
41 \( 1 + (-0.365 - 0.930i)T^{2} \)
43 \( 1 + (0.603 - 0.411i)T + (0.365 - 0.930i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (0.826 - 0.563i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (0.925 + 0.997i)T + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.290 + 1.92i)T + (-0.955 - 0.294i)T^{2} \)
79 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.733 - 0.680i)T^{2} \)
89 \( 1 + (0.733 - 0.680i)T^{2} \)
97 \( 1 + (1.61 + 0.930i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.468167114327288638640918418917, −8.223697676122238876360050496089, −7.11156373276405799972546675655, −6.61573922840521827387328455372, −5.94647030440384331094541409226, −4.76330606465897077757106883103, −4.11025628631761069561007679943, −3.29687769874291435967243641306, −2.19848002221238223474653054591, −1.43651905514544621111065399326, 1.25692908219981936707296279245, 1.86287207736807130923996757657, 3.04830946992224084923453153917, 3.95505862482543909024338425043, 4.99375276443798698008359809576, 5.67828167087610620012517400352, 6.28276537102564338457783301024, 6.99679148762186946140163700081, 7.989942808595970988040222748034, 8.456215557528429753543757027995

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