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Properties

Label 2-63e2-441.191-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 + (0.0546 − 0.139i)13-s + (0.0747 + 0.997i)16-s + (−0.623 + 1.07i)19-s + (0.623 − 0.781i)25-s + (0.826 + 0.563i)28-s + (−0.955 + 1.65i)31-s + (1.82 + 0.563i)37-s + (0.603 − 0.411i)43-s + (0.955 − 0.294i)49-s + (−0.134 + 0.0648i)52-s + (1.07 − 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 + (0.0546 − 0.139i)13-s + (0.0747 + 0.997i)16-s + (−0.623 + 1.07i)19-s + (0.623 − 0.781i)25-s + (0.826 + 0.563i)28-s + (−0.955 + 1.65i)31-s + (1.82 + 0.563i)37-s + (0.603 − 0.411i)43-s + (0.955 − 0.294i)49-s + (−0.134 + 0.0648i)52-s + (1.07 − 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: $\chi_{3969} (1808, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.851 - 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7860012819\)
\(L(\frac12)\) \(\approx\) \(0.7860012819\)
\(L(1)\) 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 \( 1 + (-0.0546 + 0.139i)T + (-0.733 - 0.680i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (-0.826 - 0.563i)T^{2} \)
31 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1.07 + 0.997i)T + (0.0747 - 0.997i)T^{2} \)
67 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.440 - 0.0663i)T + (0.955 + 0.294i)T^{2} \)
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 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.730429213002669255417178660969, −8.200778069088367991826258600819, −7.10941924609435816636583088586, −6.32975413582917249607149430702, −5.82000015717893019456894609934, −4.98958657910324268845554855011, −4.12790551183731131294948514820, −3.38600730140004207825754269492, −2.27823978081640544560916674124, −0.997200028957456888107087717002, 0.56380168775051763627398019150, 2.36663871684278969230949061974, 3.17060795403023241215387000040, 4.01056243829025064369631429157, 4.59807650756075947368658683060, 5.62741294635015265540400907716, 6.38566684335605782331120476736, 7.25980176597161752379493365583, 7.74379194210838613811375367775, 8.722025873787679069276435792628

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