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Properties

Label 2-63e2-21.11-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  + (1.67 − 0.965i)2-s + (1.36 − 2.36i)4-s − 3.34i·8-s + (0.448 + 0.258i)11-s + (−1.86 − 3.23i)16-s + 22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + 1.41i·29-s + (−3.34 − 1.93i)32-s + (−0.866 − 1.5i)37-s − 1.73·43-s + (1.22 − 0.707i)44-s + (1.36 − 2.36i)46-s + 1.93i·50-s + ⋯
L(s)  = 1  + (1.67 − 0.965i)2-s + (1.36 − 2.36i)4-s − 3.34i·8-s + (0.448 + 0.258i)11-s + (−1.86 − 3.23i)16-s + 22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + 1.41i·29-s + (−3.34 − 1.93i)32-s + (−0.866 − 1.5i)37-s − 1.73·43-s + (1.22 − 0.707i)44-s + (1.36 − 2.36i)46-s + 1.93i·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: $\chi_{3969} (3644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.511562234\)
\(L(\frac12)\) \(\approx\) \(3.511562234\)
\(L(1)\) 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 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.93iT - T^{2} \)
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

−8.581401164398414683646752260968, −7.08506756016064694232802500763, −6.89652598541036485424433152463, −5.83398983764494249437595400121, −5.21772677099297397283655976324, −4.58268342634779688040251644230, −3.68997248746991228049175165746, −3.13482913470059671358500773928, −2.11396079045775347230366327115, −1.26170016792471809735048275044, 1.89724734049271200140840222057, 3.06131283036039620208812283842, 3.57468598355903728020260799061, 4.57574743373909544824495914056, 5.01188386945216620496200311205, 5.99449968408452375661095544310, 6.42368524659529798636616346453, 7.14129505555632664750942199580, 7.907059347093015971113328033873, 8.454255166242170235873348024576

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