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Properties

Label 2-63e2-21.2-c0-0-3
Degree $2$
Conductor $3969$
Sign $0.386 + 0.922i$
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  + (0.448 + 0.258i)2-s + (−0.366 − 0.633i)4-s − 0.896i·8-s + (1.67 − 0.965i)11-s + (−0.133 + 0.232i)16-s + 22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + 1.41i·29-s + (−0.896 + 0.517i)32-s + (0.866 − 1.5i)37-s + 1.73·43-s + (−1.22 − 0.707i)44-s + (−0.366 − 0.633i)46-s − 0.517i·50-s + ⋯
L(s)  = 1  + (0.448 + 0.258i)2-s + (−0.366 − 0.633i)4-s − 0.896i·8-s + (1.67 − 0.965i)11-s + (−0.133 + 0.232i)16-s + 22-s + (−1.22 − 0.707i)23-s + (−0.5 − 0.866i)25-s + 1.41i·29-s + (−0.896 + 0.517i)32-s + (0.866 − 1.5i)37-s + 1.73·43-s + (−1.22 − 0.707i)44-s + (−0.366 − 0.633i)46-s − 0.517i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504374510\)
\(L(\frac12)\) \(\approx\) \(1.504374510\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.652234234767622826774707596320, −7.71385260679180779831576320594, −6.77078769085859391278705881177, −6.04814876549389313880089774859, −5.84110222693122679083097291198, −4.56762460389231523511090932151, −4.10489389921222743814153878414, −3.28350095563085250761790215790, −1.88663016350728270615659757650, −0.78540497855101344153888954495, 1.52455210974020000908204833182, 2.46685582316522470255571047034, 3.61208647420974965723336383348, 4.11301305331573543986297439760, 4.73844999879498036800609539507, 5.80628414742694617708912601636, 6.49127496321929678568181044086, 7.45398164831261240872900130921, 7.938762232490375973663567483751, 8.803622640613194139598316572137

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