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Properties

Label 2-63e2-441.389-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.0284 + 0.999i$
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  + (−0.222 + 0.974i)4-s + (−0.988 + 0.149i)7-s + (0.603 − 1.53i)13-s + (−0.900 − 0.433i)16-s + (−0.623 − 1.07i)19-s + (−0.988 − 0.149i)25-s + (0.0747 − 0.997i)28-s − 1.46·31-s + (1.07 − 0.997i)37-s + (−1.63 + 1.11i)43-s + (0.955 − 0.294i)49-s + (1.36 + 0.930i)52-s + (−0.425 − 1.86i)61-s + (0.623 − 0.781i)64-s + 0.149·67-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)4-s + (−0.988 + 0.149i)7-s + (0.603 − 1.53i)13-s + (−0.900 − 0.433i)16-s + (−0.623 − 1.07i)19-s + (−0.988 − 0.149i)25-s + (0.0747 − 0.997i)28-s − 1.46·31-s + (1.07 − 0.997i)37-s + (−1.63 + 1.11i)43-s + (0.955 − 0.294i)49-s + (1.36 + 0.930i)52-s + (−0.425 − 1.86i)61-s + (0.623 − 0.781i)64-s + 0.149·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.0284 + 0.999i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.0284 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5620410256\)
\(L(\frac12)\) \(\approx\) \(0.5620410256\)
\(L(1)\) 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 \( 1 + (0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.988 + 0.149i)T^{2} \)
11 \( 1 + (0.733 + 0.680i)T^{2} \)
13 \( 1 + (-0.603 + 1.53i)T + (-0.733 - 0.680i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.826 - 0.563i)T^{2} \)
31 \( 1 + 1.46T + T^{2} \)
37 \( 1 + (-1.07 + 0.997i)T + (0.0747 - 0.997i)T^{2} \)
41 \( 1 + (-0.365 - 0.930i)T^{2} \)
43 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 - 0.149T + T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2} \)
79 \( 1 - 1.91T + T^{2} \)
83 \( 1 + (0.733 - 0.680i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.318423813431723816365475003411, −7.86630748125910185536683364982, −7.03809897716107182147173465334, −6.29043860054004513289145193821, −5.54181905417636874466292087098, −4.57720209118555621285538523180, −3.59737381482700219018640810837, −3.16518491369765702970892892021, −2.21028911304560283433176797995, −0.31066957903571360051885695811, 1.41443713471656574775465933126, 2.23207445564970490958627529114, 3.68012415022423422337170019602, 4.10733271182417323973062036497, 5.17262212371983489505503136958, 6.03310816311104336376516204949, 6.43975186061662350284991720243, 7.14226788196182248284510360431, 8.193248734426534513151062544126, 9.075624991037188656060193684401

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